My friends roped me into this game a while ago and it's pretty much become a nightly affair. I now have around 15 or so champions unlocked so I usually have enough variety to keep it fresh. A few things I usually notice from new players is a lack of awareness and poor decision making. That tends to get corrected over time but the only thing I usually suggest is always err on the side of caution. It's better to let a guy escape with 5hp than it is for you to go for the tower dive and die from the turret.
Last hitting is pretty easy once you understand how quickly your champ's attack animation/damage is. But yes, as a support you should be getting very few minion kills. If you want to have some fun in a game, build a pure AD Sona. She deals out so much damage its absolutely insane.
Also if you need some help with build suggestions, check out
http://www.mobafire.com/. It's a pretty good start and some of the user guides have an excellent breakdown of the champions and their abilities.