You can get E first or Q. I like the mobility of Q personally for getting early cs or something, but it's up to the player.

If you have a kill as jax in lane. Proceed to win, learn to time his E after they blow escapes and go in for the kill with Q and E and kite around them after the stun. Phage should help you with that.

In a lane where you need to farm it out (vlad, renekton, garen etc) Just make sure to keep your distance and farm under tower, do NOT let them get a kill on you at any cost. Soak up xp and farm until level 11~13 when you have trinity force and a little for Bork. See if you can out trade them. You can? Oh yea, you're Jax. Proceed to start winning from then on.

Try to learn Jax's damage output with his W. It gets really really nuts if you can flash W for finishers and free kills. As a Jax player, and you're under tower with your E up, you should be begging for them to try and hit you. Learn to calculate your damage output under tower and make sure to remember you can activate your E early for a stun if they try to escape tower range. I can't tell you hot many sad champions i have made from using this. Oh you are going to go in on me? E ignite W block block STUN Q to minion and run away into the brush or more behind tower.
watch as your tower does the work for you and spin your lamp proud as they fall.

So if you have your third item. Use your utility to focus high priority targets like ADC's and fed people with no escapes.

The only thing you need to learn after this is to know your cooldowns and your potential to make insane plays. A Jax late game is a force to be feared. if their adc doesnt have or has their escape blown, charge in. Flash E Q Stun. Oh, their whole team is focusing you and you just 3 shot their adc without even losing half your health. Not a big deal, this should be a normal occurrence. Jax can 1v3 as long as he is not crowd controlled for any period of time. Scenario, you are getting chased from pushing tower by their Caitlyn, master yi, and swain. You see them coming because you have the map warded at all times. Any good player should have that no matter what your role. You just saw Caitlyn use her net to get closer. Big mistake on her part. Ult and EQ back in there, blocking all damage from their 2 AD's make sure to kite around swians snare, if you do it right, that's one dead Caitlyn. The Yi and swain upon realizing you are Jax, decide that is hopeless and choose to run. No big deal, ping an objective for your team while they are chased down by your Randuins active and Bork. Look at that, you got a triple kill, your team got a tower and you have 50 seconds to start knocking at their nexus.