Bridget and I have been watching Buffy the Vampire Slayer and the spinoff Angel, for the past few months. Since the shows exist within the same universe, there ends up being small amounts of crossover in the stories. You could certainly watch the shows independently and be able to follow along just fine, but watching them as they originally aired was the route that Bridget and I decided to take.
If anyone else would like to do the same as us, I’ve compiled a BtVS / Angel viewing list for your consumption. Follow this link and you will find a complete list of all the episodes in Buffy and Angel. The sheet should be self explanatory.