Since there are thousands of apps out there, from free to expensive, you may want to take a note from your friendly neighborhood librarians and do some research first. These sites can help.
Android Tapp is a good source of Android app reviews, as is Android Appstorm. Both sites provide reviews, articles, "best-of" lists, and tips for getting the most out of your Android phone.
For iOS (iPhone, iPad) devices, get app recommendations, learn of new free or discounted apps at AppAdvice or by downloading its app for $1.99.
If you're not sure what you're looking for, try Quixey, "the Search Engine for Apps."