I am a student and I have a fun problem: My friend doesn't like cats and thinks raccoon's are "way cooler." I want to build a phylogenetic tree and show him that we're more closely related to cats and therefore cats are cooler. 

However, my school assignments always told me which gene or protein to use for analyses. 

So how do I decide which gene/prot to compare without looking back at old assignments?

I'll be looking at house pets so I know I need something that cats, dogs, lizards, gold fish, and snakes have in common. 

What should be my next steps? 

Phylogenies using a single gene are not reliable. For this kind of comparison, you should probably use fast evolving genes such as mitochondrial genes. Check the literature, people have most likely already worked out which genes to use and how.