Each character only has so much you can learn before you can't improve with that character any more. Improving your game will come more from general things like team play and map awareness. You may start with 4, but I'll bet you branch out after about 10 games with each.
For tanks: Shen, Rammus, Amumu
Tanky DPS: Garen, Olaf, Xin Zhao
Melee DPS: Shaco, Tryndamere
Ranged DPS: Ashe, Tristana
Caster: Annie, Fiddlesticks,
Support: Taric, Kayle, Soraka,
Really you can just take any of the above and you'll do fine, but definitely play games with as many champs as you can.