Hey, hows it going? Welcome to the wonderful world of Saltwater Fish! This hobby is as addictive as anything, and as expensive as can be, but is far more rewarding. Especially if you are patient.
Many people run 10 gallon tanks, personally I run a 20, but a close friend has a ten gallon. They can be a real challenge, but IMO, that is what makes this fun. Do a search for Brooklyn Johnny and you will see some amazing work.
I live in an apartment as well, and I maxed out the landlords limit on gallons with 20. If you can afford it, go with a 20H. They make a great tank that is not so big that it is unmanageable, but small enough for apartment life. Plus a 20H will allow more freedom in choice of inhabitants.
If you decide to stick with a 10, I would suggest sticking with one fish. On a 10 gallon, the biofilter cannot handle very much, and fish max this out really fast. I would stick with one, if any. My friends tank had a fish, but his tank broke out with cyano and hair algae. as soon as the fish was removed, so were the algaes.
As far as fish go, as previously mentioned, a clownfish is a great addition, as would be a bangaii cardinal fish, os some of the real small gobies. A royal gramma is a colorful addition as are any of the other dottybacks.
Best of luck, and welcome aboard!