Plenty of e-readers allow you to borrow e-books from the library. In fact, I'd be safe in saying that almost all the current e-readers allow you to borrow from a public library (even the Kindle - after a bit of tweaking - which is NOT covered here...).
The steps are:
1. Find a public library and get a library card. The wiki will help you determine which library you can use (obviously, you can check your local library - then there is Philly Free Library that'll let you mail in an application to get a card for $15/year).
2. Next, you'll need a program called Adobe Digital Editions (ADE), which you can download from this site: http://www.adobe.com/products/digitaleditions/
3. After you've installed ADE, open it on your PC and follow the prompts to create an Adobe account and authorize your computer to the account (you sign up for a free account)
4. After you've authorized ADE on your computer, close ADE.
5. Plug your e-reader in and re-open ADE. It should come up with a prompt telling you your reader needs to be authorized with ADE. Follow the prompts to register your e-reader.
6. Now that that's done, go to your library's web site. Find a book (epub, PDF or mobi - depending on your device). You can follow the prompts to check out an ebook - you'll first put an e-book in your library "cart," then you'll check out (by signing in with your library card).
7. THIS IS IMPORTANT: After checking out an ebook, you'll see a button where you can download your e-book. Select the option to "open with Adobe Digital Editions"
('cause the file that'll be downloading is an ascm file, which is the ticket your computer has to have to actually get the epub/PDF/mobi file).
8. The book will download into ADE and from there, it is a simple matter to click-hold-drag the ebook to your ereader (which should be visible in the left hand column of ADE).
9. Disconnect the reader, navigate to the book (usually in a folder called "Digital Editions") and read (!).
Hope that helps.