Did you know that you can now check out books on your Kindle in some libraries? I've been not-so-patiently waiting for this to happen for awhile now (me and every other Kindle user!).
First of all, go to OverDrive Search and do a Library Search to see if your library offers this service. Then link to your library or library system to find ebooks and check them out. You should notice a format choice for Kindles and Kindle apps.
Here is a video that should help get you started. The process might be slightly different for different libraries, but I found checking out a book on my Kindle to be easy and intuitive at my public library.