Yeah, I ran the TUTS and felt pretty good about it. I used to play WC3 obsessively LONG ago, so it's sort of the same feel. Add me to your friends list, my user name is fps16
You can get Tristana for free if you like Riot's Facebook page, and Alistar if you follow Riot's youtube channel. This gives you two powerful champions that are relatively easy to use but with a high skill cap.
Don't go crazy buying 6300 IP champions unless you really want them, it is better to stick to the ones below 1350 ip, then buy an assortment of good tier 3 runes when you hit level 20.
Don't start Ranked as soon as you hit 30, you won't be ready.
A good way to step up your game is to watch High ELO streamers. You can find all of the best streamers here: http://www.own3d.tv/live