The compatibility chart over at liveaqauria will help you get started:
http://www.liveaquaria.com/general/compatibility_chart.cfm
Once you find a fish or species you are interested you can read more on them at liveaquaria.com or do a search on reefkeeping.com. I've always found great articles on fish species at reefkeeping.com.
In my experience, the fish get along fine if you keep just one type of fish in each species and as long as their body shapes are not too similar.
Tangs - I've had Koles, Purple, Hippo, and Powder Blue and have never seen any fight, not even with different types of clowns. Most people suggest if you have more than one tang, then you should try to vary their sizes, which is what I have always done.
Clowns - Some like to be by themselves, some don't mind others of the same type. But most people say not to put two types of clowns in the same tank.
Damsels - I've never had damsels, but of the people's tanks I've seen that had more than one type of damsel, I saw alot of chasing.
Chromis - I have chromis and I like them because they are very active "dither fish" meaning they swim around alot encouraging more shy fish to swim around. They are very similar to damsels, so I have not tried putting a damsel in the tank. I have seen my chromis chase each other, only to see them swimming together a few seconds later.
Gobies - You'll have to really read up on the particular goby you may want because there seems to be alot of variance in their compatibilities with other goby types. I?ve had a yellow watchman goby that got along with all everything. And now I have a green citrus goby that also gets along with everything.
Dwarf Angels - Not always 100% reef safe, and from all I?ve read, only one should be kept. I had a Coral Beauty that got along with everything.
Pseudochromis - I've never had one, so other people can jump in. But from what I've read, some pseudochromis will bully other fish species, especially if the other fish is a similar shape.
There are others; jawfish, wrasses, blennies, anthias but I don't have any experience with these. I did have a sixline wrasse that got along with everything.