Filter, definitely.
Pump... air pump? You might if your water temp is high and the fish are gasping at the surface (should not really be an issue, though).
Lighting: would be nice, though if you have enough light in the room (no direct sunlight), it isn't necessary.
Heater: only if the temperature fluctuates rapidly or gets extremely cold (which won't happen).
In a ten gallon, I'd put in one at most. However, some people on here would say 20 gallons minimum, then an extra 10 gallons per additional fish.
If you are going for just one specimen, though, I'd go with a variety that doesn't get over six inches.
However, if you want more fish or a wider selection of fish, you shoudl probably look into tropical fish instead of coldwater fish. *nods*