Hello summoners! I'm very new to the game (only level 11) and I've been trying to find the right champion for me. I like playing Evelynn but sometimes I feel like I can't hold my own in a one on one fight with her and have to gank instead to get a kill. Are there any assassin types or at least good damage dealers that I can look into?

It really depends on what you are building on her. Other good assassin sorts that are good right now are people like Talon, Zed(especially with Black Cleaver in its current state)

I was looking at Zed. He looks very strong!

Kha'zix is probably the strongest assassin overall at the moment. He'll probably be nerfed again pretty soon.

Zed, Kha'zix, Shaco, Talon, Akali are your best bets. All of them except for Zed have some form of stealth. Zed however has a very high skill cap just so you are aware.

I'm certain the closest you'll get to a very high damage assassin that can hold their own in a 1v1 situation is Akali. She has a stealth of her own, and her burst is unbelievable as well as having a fairly low skill cap. Otherwise, I hear Kha'Zix is very strong. ^^ Hope this helps

Edit: Also, ganking isn't only for your benefit of getting kills but it will help a lane that was in trouble of being zoned or lost, which will hopefully allow them to catch up and give a higher chance to win the game. As mid, I gank both lanes pretty often and sometimes even help the jungler to steal red or blue.

kat diana fizz akali. are all great assassin type that suit the style you suggested, and they can me seen in mid very commonly.