HWD,
The TCAS fit we use doesn't have the facility for the pilot to select RA, it's in permanent TA only.
Multiple TCAS conflicts are dealt with in exactly the same way as multiple visual conflicts in VFR - usually you LOOK out of the window and make a decision which avoiding action is most appropriate.
TCAS prioritises an alert on one aircraft, which may or may not be the most appropriate one, from personal experience. It may give a "false" alert if there is no mode C return. These are the frightening ones as the pilot doesn't know where to look to resolve the conflict - it could be same level, below or above him - and the azimuth may or may not be accurate - so it could be left OR right of the nose! (It might easily be up to 2700 feet above or below on our type, depending on the "airspace" setting).
I wish we could dispel the common trap of thinking that we "lucky TCAS folk" rely on it solely and fly around "heads in" because of it. We certainly don't - if anything TCAS gets a pilot's eyes OUTSIDE the window; those little white dots on the screen are a constant reminder that there are other aircraft in the vicinity!
I personally fit the TCAS instrument display into my normal lookout scan because LOOKOUT remains the primary method of collision avoidance - the ANO says so, as well as an experienced pilot's own common sense. Obviously, not all aircraft have a transponder in any event, so it would be a very unwise pilot who doesn't look out.....
As I said, TCAS isn't reliable enough in azimuth to depend on it in isolation. It IS very good in reporting altitude of other traffic, provided of course that the other traffic has a serviceable Mode C in use, so it's always best to resolve a potential conflict by altitude separation if possible.
That's why I tend to have a bee in my bonnet about permanent use of Mode C if it's fitted and paid for .... it's a potential lifesaver for both parties. If you have Mode C, please use it in the circuit too, unless ATC request otherwise because then we can "eliminate your aircraft from our enquiries". Ta.
One more point: TCAS helps a pilot to be MORE pro-active, not reactive. If you are aware of an aircraft 20 miles or more away you can obviously do something about resolving a potential conflict FAR more easily than if you are obliged to rely on lookout alone, which will result in an aircraft being "spotted" much later / closer in average conditions.
I began flying for my living in 1977 and have been doing so continuously ever since (RAF jets / Support helicopters / QHI / Heli Display pilot / Military QFI SEP / Heli SAR S.E. Asia / UK Police / VIP corporate). I've had some pretty close shaves during all that and I KNOW the limits of the human eyesight (even though I still have 6/5 vision outside the cockpit). Anything that helps me out is a bonus, as far as I'm concerned. Anyone who thinks that sole reliance on the vagaries of the human eyesight is the only good way, is sadly misguided.