Air Force ground combat units are part of Security Forces. SF units serve in two capacities -- force protection (aircraft and personnel) and policing. The force protection guys train in ground combat and could probably be used in a patrolling capacity, but they are not large units, being kept to about platoon or squad level in operation. One of my classmates is a former SF ground combat instructor.
Air Force spec ops units, in the ground operation sense, fall in three categories: CCTs, PJs, and Combat Weathermen. CCTs, or Combat Controllers, are the guys who drop in to an area and secure a landing zone for aircraft, typically behind enemy lines. PJs are the pararescue guys who are highly trained in combat (Ranger, airborne/air assault, I think SEAL training also -- unsure of the extent, but I can check online). THEN, once they are full up on the combat side, they have to basically become triage doctors. It is one of THE hardest jobs in the military because you can't just be a physical specimen since you have to know a full range of medical techniques as well. Wash out rate is astronomically high. Combat weatherman is a new position, from what I understand, but is something that has been lacking in the past. While it sounds funny, weather prediction is EXTREMELY important to special operations units in many respects. I don't know a heck of a lot about this one as I just learned of it recently.
The majority of Air Force special operations units are flying ones -- guess who ferries in all those Rangers, Special Forces and SEALs? It's not the other services using slick Blackhawks in most cases. There are other aircraft as well for resupply, aerial refueling, and psyops.