IMO, Jax is fine as he is now with the possible exception of his counter strike (IMO, the dodge or the stun needs some kind of fix <slight nerf to the dodge, or stun only affects those that have proced it>). His very nature makes him a psudo-pub-stomper (he's IMO more difficult to play well enough to pub-stomp compared to some of the more popular pub-stomper champs)
As far as easy to pick up and play champs...Ashe (ranged DPS), Annie (Mage), Yi (Melee DPS)...though IMO, you should look at all champs, find one that fits your style (Ranged DPS/Mage/Melee DPS/Tank/Support/Hybrid) and has abilities you like then hit Leaguecraft and do some experimenting (depending on your skill, opponent skill, champs farming ability, etc. you should expect to get ~10-20k gold in a SR game and probably 70-80% of that in a TT game) with item builds/runes/masteries. Most champs have decent guides to help you get started, but feel free to choose your own fate (just remember that if you go against type, you will eventually get grief from 'expert players' for your choices <they can QQ some more>)
[...][/short essay]