I might stick with the rainbows or the neons, and up the group size - 10 or 12 neon/cardinal tetras will be quite active, and with good light (set a lamp in front of the tank - the lighting int he tank hood just doesn't do them justice), they can be spectacular. Schooling fish generally do better in bigger groups, but I think they'd be ok together, too.
You might want something at the bottom of the tank, too. Corydora catfish (say, group of 5) are small, and there's a great number of species with different coloring. I have peppered corys, myself, but if you're looking for more color, panda cories (and similar black/white colored species like masked corys and skunk corys) are very nice looking.
GloFish are very colorful, but they're genetically engineered. I don't have a problem with them (as opposed to dyed fish, which I do), but some may suggest alternatives.
That said, I will, too, because I'm a shill for pearl danios. Pearl danios look pale boring pink in the store usually, but when they settle in, they're a blaze of different reds - metallic red lines on their top and belly, bright red shading on their side and tail, and they flash blue under light (set up a lamp, just like with the neons). Not quite as colorful, but I think they put on a better show than any other danio.
Anyway, there's a rule of thumb of 1 inch per gallon for stocking new tanks. There's a lot of problems with this rule, but for the fish you're looking at, it's a good safe stocking level that you won't be overstocked with.
After six months (from the end of your cycle, not initial setup), when you have some experience and are comfortable with water testing and maintainance, you can start stepping that up a bit, just don't push yourself. The more stock you have, the more attention the tank will take, and six months from now you may well have developed full blown multitank syndrome
Read up on fishless cycling at the top of the New to the Hobby section, make sure you get a good start at it and you'll enjoy the process MUCH more. Bad cycling experience and advice is probably the biggest reason people end up giving up fish. If you haven't got the tank yet, give some consideration to a slightly larger tank. 20 gallons is a good size, not too small, and you can have a good collection of fish in there. My first tank was a 29 gallon, which opened up a good few more options than a 20, and if you get a good start at the hobby (and you're in the right place for that!), you might end up wanting an upgrade before long, anyway.