Always take flash and ignite.
There are no better options.
Flash is for excaping and making sure u get the kill and
ignite is for kill securing.

If you are against an easy lane start with dorans blade or if you are against a heavy auto attack start with dorans shield and a potion.
Most of the time you want to start with rejuvination bead,pots and a ward.
Botrk is a core item in every game.It gives you so much dueling potential,you can chase with it AND you can escape really easily.Trinity force is an all around item and gives you a lot of stats
that jax needs.If they have a lot of burst buy warmogs first.If they have a fed adc buy randuins omen.Otherwise buy guardian angel.As for the last item buy either last whisper if they have a lot of armor or maw of malmortius if they have a lot of burst dmg and ap.

Normally start with q but when you invade or when you want a level 1 fight against enemy top laner then pick e.Always max q and max e second.

When you get level 6 and want to harrass the enemy jump to the enemy with q and when u have landed on him and dealt dmg to him right after use ur w with your ulti prock.IT deals a lot of damage.
Always,i mean ALWAYS buy a lot of wards not only for vision but you can jump to them.
Mid game to Late game you can 1v1 pretty puch anyone you choose unless they are extreamly fed and you happpen to fail to stun the enemy.

In mid game fights focus their mid laner mostly always and sometiems the adc because the mid laner will have a lot more damage at mid game stages.

Late game tho is different.Adcs start shining and you don't like the shine.Late game your #1 priority is to kill their adc.You can do that really well because you know...You are playing jax and jax is an epic hypercarry