MSDN's Coding4Fun has a good intro article on playing with the Flickr API using Flickr.Net API library:

"the actual mechanics of communicating with Flickr are relatively simple. The hard part is coming up with the idea for a groovy new application."

For inspiration, have play with 'the best 10 Flickr mashups', (my favorites are Tagnautica and flickrGraph), or have a browse through ProgrammableWeb's comprehensive listing of Flickr mashups.