If you purchased the books from Amazon, simply look up the book on Amazon again, and look to see if it says "Simultaneous Device Usage: Unlimited", then it is DRM free. If they specify a number, then unfortunately it has DRM
If you purchased the books from elsewhere, it is hard to say. Did the books require you to have Adobe Digital Editions to download and read the book? If that is the case, then they most likely have DRM.
The biggest thing to keep in mind is if you buy a Kindle, you're mostly stuck buying from Amazon, unless the site sells without DRM. For instance Smashwords is a great site for books from indie authors, and they sell without DRM. If you go with a Kobo, nook, or Sony reader, then you have a little more selection. The nook supports the DRM that Barnes and Noble uses, as well as the standard Adobe Adept, which is what most ePubs with DRM, or some PDFs use. If you buy a non-nook reader that supports ePub, such KoboReader, Sony Reader, etc, then you will be able to buy from everywhere pretty much with the exception of B&N and Amazon.
If the books do not have DRM, then I would recommend you to use a program called
Calibre, which organizes ebooks of all formats and lets you load them on to your device. It supports pretty much every reader under the sun. One of the coolest features of Calibre is the ability to convert books between formats. So, if you have a DRM free AZW or Mobi book (which is what the Kindle uses), and you want it in ePub, or vice versa, it will do it.
As far as removing DRM, that isn't always legal, depending on where you live. Due to that, we're not allowed to discuss it on the forum. If you really want to know, Google is your friend.