I recently got "ashe" and i can't play him and my other charcters have "free to play" on there logos so i can't play ashe? is there a membership u have to buy?

If you purchased ashe with either RP or IP she will appear in your champion pool in the pregame lobby

There is no membership to pay for, but you can pay IRL $ to get Riot Points, which can be redeamed for xp/ip boosts, champions, skin, and bundles.

If you bought ashe at the store with either IP/RP then she should show up in your champions to chose from, she will not have a "free to play" blue exclamation mark by her picture.

You don't get to use all the champions immediately. Every week (typically Monday/Tuesday) a select number of champions are able to be played without spending IP (influence points) or RP (Riot Points) on them.

League of Legends is a unique free to play game in that you can get all you need with IP (this is earned by playing games). Unless you want skins or want to boost your XP (don't) or IP, you probably don't need to spend any real world money.