Caitlyn plays somewhat similarly to Ashe, but being new I'm going to assume 6k champs are off the table for the moment.
Tristana is ok on Dominion, pretty straightforward to play, and can be obtained free (with a skin) by following Riot on Facebook. Similarly Alistair is a great tank who can be obtained for free (again with a skin) by following Riot on YouTube.
In all honesty Garen is probably the best champion to learn the game with on Dom. He's very straightforward, and reasonably effective. If you're looking for someone new though Ryze has been mentioned. He's very strong on Dom, and pretty straightforward to play. But the king of the map is Rammus. He's and absolute god on this map and very difficult to stop once you get even slightly good with him. Don't buy him if you plan on playing alot of draft as he is nearly always banned. Other possibilities for a new player include Teemo, Udyr, Poppy, Sivir, Heimer and Singed. Would have said Jax too but it might be best to wait for his rework.