If you want to keep assassin snails, you'll have to choose their tankmates wisely. As you've probably guessed by now, these are active predators that will snatch anything they can catch. Now, they're not that fast so they won't be able to harm fish and larger inverts (even bottom feeders), but care should obviously be taken when combining them with snails. Small snails are bound to get eaten sooner or later. There is a bit of ongoing discussion on whether ornamental snails like Nerites are safe, but we'd personally rather not find out.
Additionally, there is some debate on whether assassin snails are shrimp safe or not. Some hobbyists do report their assassins catching baby shrimp, although they probably don't damage your shrimp population all too much. Still, if your tank contains expensive shrimp or if you're working hard on multiplying your stock, you might want to pass up on these snails.
You probably won't be surprised to find out these snails are mainly carnivores. Their natural diet consists of snails and anything else that floats by their sandy hiding place - alive or dead. That being said, they'll also eat algae and pretty much anything they can find.
In the aquarium, your assassin snails will snack on pest snails and any leftover fish foods they can find. You can supplement their diet using commercial fish foods and frozen foods. If you're serious about assassin snail care (if you're trying to breed them, for example) you can even set up a separate snail breeding container to make sure this natural food source is always available.