Hi Janette, welcome!!
To answer your questions....
Can I get books from the Sony store if I have Cailbre?
Sure you can. You can get your books from anywhere that sells them in the format you can read on your reader. Depending on where you get them they may end up in different places on your computer but you just import them into Calibre and it will copy them to a consolidated directory (easy for back up purposes).
Someone also mentioned a problem with synchronizing, and I am not sure what that is.
I dont think either the Sony software or Calibre does a true synchronization where you can click a button and end up with the same books on your reader as your computer. I've not found that to be a problem though in that I usually just move new books to the reader. The only time it gets to be tricky is when you need to change the collection(s) a book is in. I've found it easier to delete it from the reader and then re-import it. It might work if you just copy it over - but I've not tried that way, I'm big on fresh starts
Also, there was mention of epub, which I understand what that is, but is there a problem with it and Adobe?
If you have DRMd ePub then, like toomanybooks mentioned, you'll need to download digital editions, this lets the book be opened and read and passed to your reader in a readable format.
Can you get books that are epub and then later get some books with Adobe, or there a problem having them both and you have to use only one?
As long as your reader will read them you can have as many different formats of books on it as you like.
hope that helps