So I was just thinking, if Sarada awakened the Sharingan, would she eventually have the Sharinnegan? She is Sasuke's daughter, and probably inherited some of his sage power stuff.

Well, I was wondering about the same thing. I think that since she is Sasuke's daughter, she must had inherited sasuke's six paths power, which leave us with some possibilities:
--Due to the Sage of the Six Paths' power, her sharingan awakening and evolution will be MUCH MUCH better and faster, she may be able to awaken the 3-tomoe sharingan from the beginning.
--Since her father has the EMS, and with the power of the Six Paths, she may be able to awaken an Eternal Mangekyo directly, ( Her Mangekyo pattern + her father's mangekyo pattern).
--Then, when she masters the use of her EMS and the SP Power, she'll be able to awaken the Rinnegan (Sharinnegan).