Anacharis is a good beginner plant. I have it both weighted down and also free-floating in a few of my tanks. It's a nice, cheap, edible plant a lot of fish enjoy nibbling on. My Mollies go nuts for it, so I occasionally drop some clippings from other tanks in with them. It is in all of my goldfish tanks and also in with my Endlers and dwarf platies. I rarely have to buy it any more because it grows so well in basic, stock lighting. Try plant weights, they ~help~ but you have some plant lovers there.
Other things for a newbie to try (all of which are great with fry, too, so your livebearers would have other uses for it besides just trying to eat it): water sprite, java moss, java fern, water lettuce (floating plant, just don't have it too close to the cover since the condensation can be bad for it). You may also want to try vallisneria and amazon sword.
If you ~really~ want a plant you can't get rid of, try duckweed. LOL.
I grow it (well, put it in water) in my 40 gallon pop-up pond and pop it in with my goldfish a couple times a week and they LOVE it! If I didn't have fish which ate duckweed, I wouldn't allow it near my tanks tho -- talk about nuisance plants! It's easy to contain -- you just net out what you don't want but it does keep coming back.
In the end, it all comes down to experimenting. Do you have a local aquarium society? That's a great place to get clippings to try out. You might want to post for clippings/plants on craigslist in your area to see if a local aquarist might be able to provide you with some of their excess. Your local water parameters can also affect how a plant does in your tank.
Fake plants have their place -- I use them in a couple of my tanks for various reasons (i.e., a brackish tank which does still have some acclimated java fern in it but is otherwise plastic. The mollies were pretty tough on most plants I tried before all that salt was added anyways)
Good luck!
