My first choice would be a power rack or safety squat stands, a barbell & weights, a bench.
Next a pair of adjustable dumbbells
I'd go for standard, not Olympic bar size as you can pick up the discs for next to nothing second hand.
that's your basic gym, right there. For cardio you could run, jump rope or ride a bike (outdoors to get places
). If you've the room & extra cash, then rowing, or stationary bike second hand might be an option, but in my view a luxury compared to the stuff on the first line
In a doorframe you could add a chin bar (or if you've got a power rack, problem solved as you use that for chinning (& dipping too).
You cage would be good for squats & benches in total safety, you can even get cages with pulldown attachments, but to my mind chins are superior.
There's literally 1001 exercises or more you can do with the stuff in the first sentence alone!
For bag work I've got a big 5ft hanging bag, a mid-sized kickback, that can be hang or held & a couple of precision bags for when mates want to come & hold them. I did box many
years ago, but I'm no real expert as I never took it as a serious sport, just a bit of fun, but I expect any boxers (or MA people) can point you towards some good equipment