I have a 100 amp service run from my main 200 amp service at house run to pool building. I want to run electric from pool building underground to a detached garage. I'm wondering what size wire to run and what amp panel in garage. Garage would use some lights,door opener and run 120 v equipment such as air compressor,flux welder, small propane reznor heater and table saw, typically not all at the same time
You haven't stated much about the subpanel in the pool building, and it would be nice to have 100A in the garage given the plans you have, and the plans automakers have for future vehicles.
So I would simply extend the pool circuit as 100A. You can do that with lugging only - you won't need to add a bulky circuit breaker in the potentially small pool panel, since the 100A main panel breaker feeding the pool will also protect the run to the garage, being all 100A.
100A feeder is typically #1 Aluminum, unless you own a copper mine, in which case #3 copper will suffice. I'm not a fan of copper because most splice lugs and subpanel lugs are zinc plated aluminum, and why introduce dissimilar metals and pay more for the privilege?
On the garage panel, I see a need for at least 14 spaces (assuming you graduate to 240V compressor, table saw and welder, plus 4 120V circuits plus two 240V EV chargers. The worst mistake you can make is short yourself on breaker spaces, so think big - like 30 spaces. You do need a shutoff switch in the garage subpanel, and usually people just select main-breaker panels but the amp rating doesn't matter. 30 space panels typically have 150-200A main breakers, which is fine.
Never rely on 'double-stuff' breakers to solve full-panel problems. Most breakers these days need to be AFCI and/or GFCI, and those are simply not available in double-stuff.
It would sure be nice if the pool panel had "thru lugs" for onward connection to the garage. Think about changing the pool panel for one with thru lugs.