All right, clown gobies unless its a mated pair WILL fight in that small of a tank.
I've been accused of misleading people into overstocking their nanos so here goes.
I don't recommend keeping clowns in a 10gallon, but people do it. I'd do only TRUE nanofish in a 10gallon.
I promise you that if you follow these guidelines that you will have a cool looking tank on a budget.
You will have to scout your local forums and classifieds on here to make sure you get the lowest possible prices, but with $50 a week, you could probably get a $10-$15 dollar fish pretty easily.
I'm let the tank cycle really well since you don't know how cured/cycle that rock is that you're gonna throw in there and the sand too..
What I'd personally do are the following AFTER the tank is cycled and showing some hair algae on the rocks:
Throw in 5 astrea snails, 3 big variety nassarius or 6 small ones. That's it for the cleaning crew. If you can afford to do so at the same time, pick up a candy stripe (aka randall's) pistol shrimp. Then a week after getting the shrimp, get you a goby to pair up with it such as a hifin red banded (one of the cheaper ones) or a yashia (I've seen them for as low as $30 at a LFS, they're also REALLY colorful and my favorite) which you could probably find for cheap on the classifieds here. Then let the tank run for awhile and mature for a month or two while you're saving up... You can throw softies like zoanthids and ricordeas in there in the mean time to give it some flavor. I would personally recommend AGAINST any sort of xenia, kenya tree, clove polyps, or green star polyps as they'll overgrow a 10gallon in no time.
After the tank has been cycled for 2months or more, get you 2 neon gobies and a rainfords goby. Then I'd recommend that you call it a day for fish or the maintenance will get exponentially greater. You could probably get some cool LPS in there at this point like acans, closed brains, chalices, etc.