I'm trying to find a list of all of the plant species. Is there one? I don't want to use a mod, but want to keep track of the plants I get. Also, on the breeding charts, just to clarify something for myself, is it possible to have every type of flower on each type of foliage? Thanks in advance!

The answer to both of your questions is "YES!" There are many different lists and charts that people have developed. A lot of them can be found easily in the Plant Tycoon Guides at the bottom of the main forum page.

There are 23 types of foliage and each one may have the 23 types of flower who exist! It's why there are 23 x 23 = 529 plants. I don't think there is a list but the charts may show it.
Like this:

Thank you both so much! I got to looking around and discovered Borg's charts and his virtual seed tray! I have to say, he has my undying gratitude!!! The seed tray even hides the lineage so it doesn't spoil it!

Good!
But take care to never open the VST and the PT Game in the same time!! or you lose all your seeds!!