Live plants are actually negatively affected by ammonia, at concentrations over 1ppm they can effectively melt creating higher levels of ammonia in your tank and further jeopardizing the health of your fish. You should not have any ammonia in your tank, and if you do your tank may be either cycling or overstocked/overfed. its also important to note that plants exposed to ammonia at higher pH levels (7+) will have a greater negative impact on the health of the plants. If your tank is not cycled you should see the many tips and articles about cycling your tank within the forums here.
That being said, most plants will take up some excess "nutrients" i.e. nitrate and ammonium (*not ammonia*) but they won't completely remove them. You will still have to do water changes.
I'm not sure about the diatoms I've never heard about plants helping to remove them. but someone might have more extensive knowledge about this.
I'd state the three floating easiest plants for you to grow would be water lettuce, duckweed, or amazon frogbit. As for places to purchase them if your local fish store (LFS) has a shrimp tank they will likely have at least some of one of these kinds of plants as shrimp love to clean their roots of biofilm and algae. You can also collect duckweed from a pond but be wary that it can contain things you don't want in your aquarium.
The easiest way to remove pests and snails is using a bleach water solution and you just dip the plants in for 5-10 seconds and it should remove any unwanted clingers.
Another thing to mention is that plants like duckweed are quite voracious in their growth rate and can quickly block out the entire surface of the water so it is important to either contain it or to remove it as necessary. If you have other plants they will require light too.
I would highly recommend waiting until you have 0 ammonia in your tank before adding plants. You don't want to end up spending money on plants only to have them killed by ammonia present in your water (been there, done that....)!!!
