First of all, my tip. Pay attention to the other team's composition, and factor it into your build. Are they 4 physical carries and a Cho'Gath? Sweet, just have everyone, even the purest DPS carries, buy Thornmail. Are they caster carries? Banshee's and Merc's for everyone, no exceptions. You'll be surprised how many players, even decent ones, forget to do this. To make sure people don't forget, just make a friendly note of it during the start of the match. Say something like "Lol all their carries are physical, let's buy Thornmail".
As the above poster said, Ashe and Annie are great to start out with, and still do well at higher levels of play. Just watch out for ganks, since they're quite squishy.
Garen is a great Melee DPS to learn on, and he excels early game.
For tanks...well, leave tanks alone until you're playing alongside a competent team that will follow you in after you initiate.