Books on finite languages and automata theory with examples, problems and solutions
Try this question to get an idea. I think the classical ones at universities are the one from Sipser and Hopcroft's one. For Hopcroft I would really look for the old version if you are mathematically inclined. They removed a lot of proofs in new versions. 

For specific examples of problems you'll get solutions here and there over the Internet. It is good not to have a straight look to the solution and having the solution very near can be very tempting at times. 

While looking for books to get an idea of what's on them I like this Internet library. Then you can always buy the book on paper or borrow it from your University library.

Hope this helps.

Some books with solutions: (under construction)

Hopcroft's et alii. solutions to their book. 

Apparently, Sipser's book contains solutions after each chapter.