Do I rush TF?

If i do, How do I heal in fights? Everytime I see jax, I see them heal for an insane amount in fights early on even.

rush twisted fate

I think it's still fine building hybrid on Jax.
Besides, Trinity Force got nerfed

Hextech Gunblade

Rush Tri-force first.
I start with Q at level 1, E at level 2, and Max W after that
After Tri-force is up to you, most people either go:
Gunblade (has spell vamp and life steal so more sustain, plus its active is nice)
Guardian Angel (more armor and MR plus the revive)
Rageblade (similar to gunblade)

Forget Trinity. Mostly you just want a ton of AP and AD and some form of chasing power.

Iceborn Gauntlet works wonders.

The insane healing you're seeing is from Gunblade. Every single one of his spells procs it, even his ult's third strike bonus.
His recommended build should give you a pretty solid idea of the order in which you build things. After playing him enough you'll get an idea as to what to buy in what situation.

I find Twisted Fate to be pretty squishy against Jax. I mainly use a jungle build on Jax in lane and I have never had TF do much damage on me. :-)

if you are dominating your lane. gunblade will make u snowball faster. if ur not doing well just get vamp scepter and tri force.
can get ga after 1 full item or after gunblade and tf. ga should be 3rd at latest.
personally i dont like rageblade on jax. wits end is good. the new item (cdr/attackspeed/ad/tenacity) cant remember the name right now. last wisper or BC if they are rly tanky. warmog is nice sometimes too if they all building penetration.