Mainly furballing with historic plane sets. As you say, if they are running a Battle of Britain set, it is Spits / Hurricanes against 109s, 110s and JU88s -- no Zekes or Corsair. They will sometimes make a substitution in an attempt to fill a gap in the plane set (if one exists), but mostly stick to "history" when possible. Usually this means fair / balanced plane sets and lots of fun fights.
Occasionally someone will set up a mission or objective that may involve a base capture, but for the most part that is left for the main arenas.
The main arenas are separated by time periods (early, mid, and late war), but are free-form from there. That is, you can up any plane from that time period or earlier, regardless of what side you are on. So you get P51 vs. Spitfires vs. LA-7 action in there. There is also a base capture / "win the war" aspect to it that some like, others don't. Plenty of room to find what you like and do that (jabo; bombers like B-17s, B-24s, Lancasters; furball only; tank fights, etc).
You can form or join a squad in any arena. Although there are separate Mains, the Main Arena squads can fly in any Main and remain affiliated. AvA squads can be separate if you wish. You can move freely between the arenas any time you want -- there is no restriction or having to "choose" an arena for any length of time.
The game comes with a free two week trial where you can play in any of the main arenas or AvA. I strongly suggest downloading and practicing offline for a while before starting your two week free trial. You may have been playing another flight game, but as I am sure you are aware, there are likely going to be large differences in the flight models and how the planes handle. Get used to flying and shooting drones offline first to minimize frustration and maximize fun time when you do your free trial. When you start, you may find a short amount of time in the Training Arena helpful as well -- there are almost always volunteer trainers in there to help with any questions.
Hope that covers everything you wanted to know.
and welcome!