The internet is a great resource for readers, and I take advantage of it as often as I can. Copyright Law is a complex animal, but basically, anything published before 1923 is no longer in copyright in America--and many later works never had their copyright renewed. In countries like Canada and Australia, copyright does not extend as far back as it does in America. And whenever works do go out of copyright, there are armies of fans, readers, and academics out there who take the time to scan, edit, and put them up online for our enjoyment, readable on tablets, ereaders, or plain old computer screens. Here are a few of the resources available online for free, public domain ebooks.
Project Gutenberg: Made up of a large group of volunteers, Gutenberg hosts tens of thousands of free books in various formats, including Kindle, PDF, and plain text. You can usually find any notable text written in English from before about 1925, as well as some more modern ones that are no longer in copyright. Project Gutenberg Australia has books from 1955 or earlier. They don't have as many books as the American site, but if you want something newer, they are a good resource.
Wm. Caxton from The Graphic, 1877
The Internet Archive contains scanned books from hundreds of university libraries from around the world. What's remarkable is that, for most of their books, you get to see the original formatting, illustrations, and other such details. First, click on the book icon to the left of the page, then click on 'eBooks and Texts', and finally, search for books in the search bar on the far right where it says 'Search this collection'. Once the results come in, you can click on any book you see and even read it online by clicking through the pages at the top.
However, I've found that for many of their books, while the PDF version will be fine, the other formats will often have problems with formatting and spelling as a result of their computers trying to turn pictures of book pages into words and letters. They also put up other types of files, such as music and videos.
There is also Google Books, which lists pretty much every book ever written--though only some of them are available for free. To find the free ones, search for an author or title, then press enter. When the results come up, click on 'search tools', then 'any books', then select 'free Google eBooks', and it will show you a list of free books related to the author or title you searched for. Once you've selected the book you want, go to the right of the screen and click on the little gear, which will give you the options do download it as a .pdf, .epub, or plain text. You can often read more recent articles and collections this way, even if it doesn't let you download them as eBooks.
Kipling - With The Night Mail
Also, if you're looking for a book by a specific author, sometimes you can check out their Wikipedia page, then scroll down to the 'external links' section, which often gives a list of pages where the author's works can be found. For example, here are the links on Chekhov's page, which actually go to his lists on Internet Archive and Project Gutenberg, but which often give you other options, such as fan-pages for that author which provide books you can't find elsewhere.
Amazon also has a selection of free eBooks available for Kindle, though if you get the computer program Calibre, you can change Kindle files into any other type of eBook file (and vice-versa). You can also check out ManyBooks and OpenLibrary. I know they have thousands of free books available, but I haven't used those sites yet.
Formatting and Reading
Once you've found a book you want to download and read, you have to decide what format you need. The Amazon Kindle uses the .mobi format, as well as its own format, .azw and .azw3. The Barnes and Noble Nook uses the industry standard .epub format. Both readers, as well as tablets and computers can usually read .pdf files, too. One of the simplest formats is .txt, which pretty much any machine can display for you.