The airshows change a bit every year depending on which aircraft are flying, how many pilots are available and other things. Yes, there is a dogfight re-enactment with at least two and as many as four aircraft. There are also bombing runs, barnstorming competitions including balloon-busting and the Delsey Dive (I'm not going to spoil that one for you, you'll have to go see it), sightseeing rides in a 1929 New Standard and aerobatic demonstrations.
Historically, the Saturday show has been a "History of Flight" theme, showcasing very early Pioneer-era aircraft, a brief WWI act and lots of Golden Age machines. Sunday was always the WWI melodrama, although many of the Golden Age birds were part of that act as well. Last year the show was the same both days; again this depends on aircraft, pilot and performer availability. You might want to read through the "
Cole Palen's Old Rhinebeck Aerodrome" thread, which contains lots of great pictures and stories and will give you some idea what the airshows are like.
Honestly, even IF they "just flew around," you should never pass up the chance to see an original Bleriot struggle into the air, watch a Jenny turn lazy slow-motion circles overhead or smell the castor oil of a real rotary engine. Go.