Here I will cover some of the basic lane opponents that you might go against and their difficulty.
Your life will be hard before you hit 6 so just play passively and farm until you get to 6. When you fight him make sure to use ignite to drastically decrease his healing from his w. His passive is really annoying so you may have to retreat if his passive is up after you have triggered it. If he initiates with his q he has no way to dodge your [skills/jax/e.png] so its pretty much gg for him.
Darius makes it really hard to cs with his spinning axe of death. You have to play really passive and ask your jungler for a gank to get ahead. Whatever you do try not to let him get passive stacks on you because when he presses r with full stacks you are doomed.
Don't really see him top lane very much. You want to buy some armor pen because he will most likely build some armor against you. Watch out for his crazy health regen. He can get really hard to kill when he gets tanky. Ask your jungler for ganks.
She just got nerfed so she's not as big of a threat as she used to be. Wait for her to use rappel before you use [skills/jax/e.png]. Stay behind minions to dodge her poke.
She will be annoying until you hit 6. Then just use your combo and if she commits wait until she uses her ult to activate [skills/jax/e.png] because it completely mitigates all of the damage from her ult. You probably won't even need ganks from your jungler.
Naturally Tanky. His silence and spin to win combo can do a lot of damage if he gets ahead and once he starts to get tanky he will be almost impossible to kill. Ask for a lot of ganks.
Use your combo. Whoever activates [skills/jax/e.png] first in a 1v1 loses. Activate it right after he activates his and you will win.
Can poke from a great distance. His knock back completely makes your [skills/jax/e.png] useless. Ask for a lot of ganks and try not to go all in with him unless you are sure you can win.
Avoid his poke and stuns. He is very squishy so your combo can do a lot of damage to him.
Really tanky. All of his debuffs really destroy auto attack based champions like you. Use your combo a lot and ask for ganks.
Even after his rework he is still easy to beat. He is an auto attack based champion so your [skills/jax/e.png] mitigates all of his damage. Sometimes it is wise to wait for him to use his meditate before using your [skills/jax/e.png] Late Game he will be a pain in the ass but you scale into late game way better than he does so you will destroy him.
His high sustain is annoying. His wither just destroys you. If he uses wither do not attack him until after it has worn off. Be careful about his ult. After he uses his Q he really doesn't have anything else until it gets off cd so remember that when fighting him. He can get tanky fast so also be cautious about that. NEVER LET NASUS FARM! Try to deny him farm as much as you can and his q will be useless.
This guy is a bitch to deal with. You are a Late Game Champ. He is an Early Game Champ. His poke is annoying as hell and his full combo can be devastating. Make sure to build a lot of armor against him. Late Game he stands no chance against you so just wait until then
She has one of the strongest early games out of all the top laners and since your early game is weak she can dish out a lot of damage. Play passively and ask for a lot of ganks because she can be really hard to kill. Late Game she stands no chance against you so go out there and kick her ass late game.
His poison trail makes it very difficult for you to do damage to him without getting poisoned. Ask for some ganks. He can get really tanky really quickly so try to take him down early.
He has range over you and his blinding dart does a lot of damage while also preventing you from damaging him with your auto attacks. He is very squishy so your combo can do a lot of damage to him. Also, don't go in the bushes. That's where he puts all of his shrooms so running around in there like an idiot won't help you. Oracles completely destroys him and it only costs 400 gold so you should consider buying it early on.
He is an auto attack based champion so you can easily beat him. His ult is annoying. Make sure to ignite him about halfway through his ult to get the kill on him. Shouldn't be too hard to beat him.
He is an auto attack based champion. He has an insane early game so play passively until you hit 6. Then when he uses his combo you will most likely smash his face in since he has no way of escaping after he dashes.
This guy is the king of zoning. He will make your life living hell in lane while being fairly tanky. Try to farm as best you can and ask for ganks from your jungler a lot.