Landsknechts are a somewhat-generic damage class. Their primary job is to hit things. That's really about it. A lot of their methods of hitting things have extra effects attached, but at the end of the day, their thing is hitting things and being more durable than most other damage dealer classes. They have two primary skill trees: swords and axes. Swords are reasonably accurate and quick, while axes are somewhat innaccurate and slow, yet generally hit harder than sword skills. In addition to their attack skills, Landsknechts have access to a few pseudo-support skills, of varying usefulness.

Stats

(Full stat table)

Landsknechts have pretty good STR, although they're outdone a little in that department by Highlanders and quite a lot by Ronin. Their TEC is pretty atrocious, giving them low defense against TEC-based attacks and pretty bad ailment/bind resistance, despite their decent LUC. Their AGI is similarly bad, which leads to them being really slow in the turn order (especially when they're using axes), and adversely affects their accuracy. Their HP and VIT are both very, very good, which gives them a pretty good amount of survivaibility compared to other damage classes. Lastly, their TP is pretty bad--not much else to say on that front.

Common Passives: HP Up, TP Up, ATK Up, DEF Up
Gathering Skill: Mine


Sword Mastery
Requirements: None


Needed to learn sword skills. Increases damage dealt with swords.
Swords seem a bit underwhelming at first, but with some careful application they can prove to be quite a reliable tool in a pretty decent spread of situations. They won't pull as much damage as, say, a Ronin, but there is an appeal in having zero setup time to fire things off, which can prove to be a life-or-death advantage against some nastier obstacles.

Raging Edge


Deals melee STR-based cut damage to one enemy. Has no speed modifier at all levels.
As basic as skills get. Deals a pretty nice amount of damage for how much TP it uses, surprisingly.
This is pretty much the bread and butter of a sword-based Landsknecht. The main thing of note here is the large jump in damage and TP cost at level 10; This is fairly exploitable with a Boost alongside a decent-leveled Charge effect in the early-midgame to get some pretty hefty burst damage without the TP cost dragging you down.

Tornado


Deals melee STR-based cut damage to one enemy. Deals splash damage to adjacent targets. Has an 80% speed modifier and a base accuracy of 98% at all levels.
A bit of crowd control in random encounters--emphasis on "a bit." It's not as good as clearing out random encounters as, say, an Alchemist all-target Formula, but then again, what do you really expect for an early sword skill?
Tornado's... in a bit of an odd place for EOU's general viability. Its main draw is its cheapness in exchange for being a splash attack, but despite the lower cost it falls into a bit of a situation of being an all-around type skill instead of specializing in single-target elimination or a mobbing nuke. It's okay for mobbing, especially if you're dealing with very large crowds and you need a specific target eliminated, but for that purpose I'd lean more towards Raging Edge. If I need a crowd cleared out as much as possible, though, there's always Falcon Slash.

Blinding Slap


Deals melee STR-based cut damage to one enemy. If the attack misses, attempts to inflict blind on all enemies. Has a 50% speed modifier at all levels.
An odd duck, as far as skills go. The blind part is absolute garbage, due to Landsknechts having bad TEC and LUC. The damage might seem nice on the surface, until you remember that not only are you gonna need a disable for setup (either blind, paralysis, panic, sleep, fear, or a leg bind), a level 10 Raging Edge deals 50% less damage for 14 less TP, and requires far less setup to actually be reliable.
Oh, Blinding Slap... There's not much to say here beyond what Ragnar's said, but one thing to note that there is a slight mitigation built into Landsknecht's skill set for the accuracy penalty in Triple Charge. However, it isn't quite enough to make up for the massive accuracy penalties at later levels. This is probably the least appealing skill in Sword's toolkit.

Falcon Slash


Deals multiple instances of melee STR-based cut damage to random enemies. Can hit the same target once at most. Has a 120% speed modifier and a base accuracy of 98% at all levels.
Falcon Slash is pertty expensive relative to Landsknecht TP pools--but, on the other hand, it's their best form of clearing out random encounters. It's not something to rush, pretty obviously, but it's still plenty useful.
That TP cost is scary at first glance, but with the overall layout of EOU's systems, this is a much more appealing skill compared to its rather poor showing in the sequel. With the combination of Amritas being available as soon as B11F and readily available fountains, Falcon Slash goes from being an inefficent resource hog to a very powerful tool for decisive sweeps in nastier random encounters. The cost does add up quickly, however, so be careful about spamming it. Unfortunately, with the TP solutions that exist, this does undermine Tornado's appeal of being efficient.

Axe Mastery


Needed to learn axe skills. Increases damage dealt with axes.
Now here's where it starts getting weird. Axes have a lot of strange skills that don't quite suit the stat spread of a Landsknecht and can be somewhat unreliable. In my opinion, however, Axes have a better use elsewhere...

Boomerang Axe


Deals ranged STR-based bash damage to one enemy.
Deals 70% less damage, at level 10, than a Raging Edge of the same level, for just a little less TP. I dunno about you, but being ranged doesn't make up for that, in my opinion.
Not much to say here. There's a potential application in ignoring rude front row/back row denial setups, but I've honestly been able to get by with just swords in those situations anyways.

Head Bash


Deals melee STR-based bash damage to one enemy. Attempts to bind the head of the target. Has an 80% speed modifier and a base accuracy of 98% at all levels.
Still deals mediocre damage, and like I said all the way back at the start, Landsknechts having a bad TEC/LUC spread means that head bind chance means almost nothing.
Yet another poor showing from Axe's base skill tree. This is pretty much junk and a job infinitely more suited to a Dark Hunter or a Hexer. This might have been better if the base chances were a little higher, but not every random disable skill can be Disabling Shot, I suppose.

Double Strike


Gives the user a chance to attack twice when using the Attack command.
Uh.
Unfortunately, to my knowledge at least, this doesn't apply to chaser effects. Which is a bit disappointing since that would make them a bit more appealing.

Stunning Smash


Deals melee STR-based bash damage to one enemy. Attempts to stun the target. Has a base accuracy of 98% at all levels.
See my comments on Head Bash. It deals more damage and becomes pretty fast, I guess.
It does a bit more damage than the last two Axe skills, I suppose? 292% at level 10 is still a pretty poor showing for something that costs 19 TP for a single target attack. Although, honestly, the speed modifier doesn't appeal to me that much. Axe's speed penalties combined with Landsknecht's low AGI pretty much ensure that they'll probably get outsped regardless. And since this is a Stun effect, that essentially makes the status chance useless.

Fire/Ice/Volt Chaser


For one turn, follows up on any fire/ice/volt attacks with melee STR-based damage. The element is either cut for swords, or bash for axes, plus the element of whatever was chased. Each time an attack is chased, the chance of the user following up on another attack on that turn is decreased. The initial follow up chance is 100%. There is a limit on how many times the user can chase attacks per turn. Initial prep has priority. Chase hits have a base accuracy of 98%.
Chasers are arguably the signature skills of Landsknechts, and, erm... They're something of a mixed bag. On the one hand, in EOU, you have the benefit of Action Boost existing (in an NG+ from Story mode if you're not a save-editing trickster), meaning you can get an easy three chases just from one character. On the other hand, without that, you'll need at least three chases for Chasers to be worth using over something like Raging Edge or Power Crush, and that can be pretty tricky to set up.
And here is where I think Axes start to shine a bit more. This is where Landsknecht starts to get weird. At first glance, these skills are absolute trash. They don't do much damage for the effort it takes to actually set up a functioning party to use them, and the maximum chases seems impossible to reach. However, there are some skills laying around that can help...

Power Crush


Deals melee STR-based bash damage to one enemy. Has an 80% speed modifier and a base accuracy of 98% at all levels.
Costs a lot of TP for just slightly more damage than a Raging Edge. I know that doesn't really account for the difference in ATK between swords and axes, but that doesn't really change much.
If you aren't going for chase setups with Axes, this exists, I guess? It costs quite a lot for not much more throughput when compared to Raging Edge. But if you absolutely insist on doing conventional attacking with Axes, then this is what you should be using.

Triple Charge


Increases the damage, accuracy, and disable infliction chance of the user's next attack. Has priority.
Pretty much designed for use with Blinding Slap, which... Well, that's not the best use of TP relative to damage. A level 10 Blinding Slash + Triple Charge is 595% damage for 30 total TP, while two Raging Edges in a row are 640% damage for 12 TP.
It's a charge skill! It has a net loss of turn economy! It doesn't really do anything for Landsknecht outside of Blinding Slap! For raw damage with far more reliable skills with Boost setups, I'd pick up a Charge grimoire from any of the multitude of enemies that can produce one or Action Boost if you're nasty. Triple Charge does have some applications for Dark Hunters and Hexers with the boosted infliction rate, though.
War Cry


Targets the user and the row they're in. Consumes a percentage of affected party members' current HP at cast time. Increases attack and lowers defense for a set amount of turns. Has an 80% speed modifier.
I've seen a lot of people talk about the virtues of Bloody Offense on Highlander--which is entirely justified because it's a good buff. However, Highlanders are (normally) exclusive to Story mode. War Cry, on the other hand, is plenty accessible in a normal Classic save, and gives the exact same buff amount at each level. The downsides are different, yeah, but they're both similarly negligible--a 20% decrease in defense is about as dangerous as a 15% current HP drain on every action are both equal in terms of danger, in my eyes.
This was a hell of a sleeper hit in Landsknecht's tool kit. It is a very powerful attack buff, and with another attacker on the front row like a Dark Hunter or a Ronin, you can set up for some serious damage output. A highly recommended skill for Landsknecht.
Recover


Restores a percentage of the user's maximum HP. Removes binds from the user. From levels 1 to 9, if the user has more binds than Recover removes, it will remove random binds. Has an 80% speed modifier at all levels.
I don't really see the value in investing 13 skill points to max out Recover.
SP hunger aside, this is a slow healing skill on a slow class with fairly limited utility. Leave healing or removing disables to a Medic or a Survivalist.

Savage Cry


Increases the user's chance of being targeted by enemies, and reduces all enemies' physical defense for a set amount of turns. Has no speed modifier.
Not a bad defense debuff, and Landsknechts are durable enough that they can probably be okay with the increased target chance. Protectors can help with that second part, too.
I wouldn't really suggest this on a Landsknecht proper. This is definitely something to consider for a Protector via a Grimoire Stone, if you're okay with regaining levels after resting to get it off of your Landsknecht afterwards for better SP allocation.

Strike Chaser


When the user next damages an enemy, all other party members will attempt to attack the user's target with their equipped weapon. Chase damage is increased if the attack being chased is of a different element than what the user hits for. Each time an attack is chased, the chance of another attack is decreased. The initial follow up chance is 100%. Initial prep has priority. Chase hits have a base accuracy of 98%.
A really weird Chaser skill that takes some cues from Shoguns from EO3. Its best use is if you have an Axe Landsknecht and have at least two other front row party members that, most likely, have weapons that deal cut damage. At level 10, that works out to 818% damage total (which, to be fair, is dependent on the attacking party members' STR stats), and at 10 + Boost, it works out to 1140%.
And here's one of the aforementioned skills in my comments on the elemental Chasers. This skill is exceptionally odd, but once you set it off with the first elemental hit for a chase to fire off, your allies can set off even more chases by following up your initial chase hit with elemental weapon buffs from a Troubadour or oils. This pairs up well rather nicely with Perfect Chaser on a Ronin, another strange followup skill that is essentially the inverse of this one. The applications of Strike Chaser for conventional attacking are questionable, mostly due to Swords being the standard attacking weapon of choice as well as being Cut, which is shared with a lot of frontline weapons that you'd want to be using Strike Chaser with.