Irelia's laning style is simple: You farm passively with auto attacks and Bladesurge until you get a few levels on Hiten Style. You should be able to get most, if not all, the creeps.
After that you should start trading, the true damage and health gain should keep you ahead in trades. Careful! Once her skills are on cooldown, Irelia is very weak as a fighter - unless you have a Phage and/or Wit's End.
Everytime you want to recall, use Trancedent Blades to push the lane as to not lose creeps while shopping.
On fights you should initiate when you see someone off position and abssorb some damage and look out for a vunerable carry. Jump on them, stun them and deal as much damage as you can. Because of her passive, she can go in and out of the fight with ease, getting health back when she isn't the focus with Hiten Style and Transcendent Blades.
Finally, if you manage to have their carry run away, most times it would be better to go back and peel for your teammates, unless finishing the carry off will only take a couple seconds. With Hiten Style active, even tanks will go down fast and you really want your carries alive.
As the initiator you are expected to take lots of damage, thats why you build tanky. If it seems you ll go down, try to delay that as much as possible by attacking back and gaining health so your team will have a chance to clean up while the enemies bother with you. But if you can go out on low health and re-engage when it's safe, aim for that.
Tips & Tricks
Irelia has some important techniques players should never forget.
The most common one would be double Bladesurge. This is accomplished by using it on a dying minion (or a full health ranged one, provided you have a sheen proc and Hiten Style active) so it comes off cooldown, then jumping again on the enemy. In the laning phase this is used to harass the opponents by killing the minion then starting to attack them.
Which brings us to the next tip: Always try to use Equilibrium Strike when you are on lower health or even than the opponent so you get the stun. Side note: This skill allows for very good baits, especially under turrets.
And following it up with: Bladesurge allows for a second auto attack. By hitting then once, you Bladesurge on them, the attack animation ends and you hit them again. This results in a triple hit, which translates in a lot of burst damage and health gained back, when Hiten Style is active.
The maximum damage on the lane phase would be: Bladesurge on a minion, Equilibrium Stike them (best results when you're on lower hp), activate Hiten Style and start hitting. When Hitten Style is almost over or the opponent is outside your attack range, Bladesurge them and get the free extra auto attack.
Irelia doesn't really get much stronger with specific teammates, the reason she is picked is that she can work with or against anyone.
She's fine with champions that have shields, as they let her stay in the fight longer, as with heroes with burst damage, because she can go in and pick up the kills after the first cooldowns are blown.
Some notable match-ups:
Yorick: Try to farm against him, use the brush to avoid his ghouls and ask for a gank, as he can die quite easily. You should be able to take him on by yourself with the right items and constant pressure. Build Doran's Shield into Merc Treads, then Phage into Wit's End. Buy lots of potions.
Malphite: Try making him waste his mana as you regen your health. Build an early Sheen into Wit's End. If he uses his E, then try to punish him with Hiten Style.
Olaf: Focus on farming and avoiding his axes. He must not start chaining axes while chasing you! Aside that, you can Equilibruim Stike him into some Hiten Style hits after he uses E on you, with care. If you can get a gank early, before 6, thats good. After that he should be ungankable so the goal is outscaling him on fights.
Jax: He should try to harass you with W. Stun him and get some hits in with Hiten Style. He should use his E. Wait till its almost over and then jump on a creep with Bladesurge to avoid the stun.
If he ever uses E to offensively harass you, punish him with Hiten Style after it's over. You should ask for ganks here as Jax is a monster late game.
Jayce: He will annoy you a lot and will try to keep you from farming. Avoid his Q and stay mostly in the brush. You can trade with him with a Bladesurge on minion and stun, but he will knock you back after a couple hits. Get Ninja Tabi and Doran's Shield asap.