Question

How do animal like protist grow, survive, and reproduce?

Answer 1

First, a protist is not an animal.
It's a group of unicellular eukaryotes.

Nutrition for growth is different for every kind of protist, some use filter feeding and some eat bacteria.
Reproduction is sexual (conjugation) or asexual (binary fission)
You might want to look at http://en.wikipedia.org/wiki/Protist for the full story :D