What you have now is a 3-piece ("cup and cone") square taper BB. It definitely has english threads and the width of the BB shell is either 68mm or 73mm. To replace, you would also need to know the length of the spindle. Only accurate way to know is to pull the cranks and measure.
The typical "upgrade" on this would be to go with a standard, mid-range Shimano cartridge BB (UN54/55). There's no reason to get anything expensive or exotic.
That all said, unless your current BB is worn/damaged, there is no reason to upgrade it. 3-piece BB's actually perform as well as newer cartridge BB's as long as they are maintained and kept in adjustment. Cartridges' big advantage is that they are sealed and don't need regular cleaning or adjustment. If you need to replace the current BB because it's worn, I'd go with a cartridge type. But otherwise, swapping it out won't get you anything.
If you're changing cranksets, you may need a new BB because it calls for a different length spindle. But not necessarily.
In general terms, here's my advice on "upgrading". It's usually not going to get your much unless you are replacing a part that isn't functioning or is of particularly lower quality than everything else on the bike. Because so many of the parts on bikes work together, it's hard to get much of a performance upgrade without changing all the parts in a particular assembly (like cranks, derailleurs, shifters, chain, .......)
My exceptions to that rule are tires, brake pads, and seats. These things will make a big difference in the bike's performance regardless of anything else.
But of course, YMMV; carry on as you see fit.
awesome reply, thanks, a lot of wisdom there. I had not considered keeping this BB and just replacing the crankset, but I see that is an option.
The bike is somewhat heavy, and I want lower gearing, probably a 22/32/42 or at least 24/34/44.
I'd like to lighten the weight, but no obvious way except for the front fork, and I'm not sure that I want to give that up. My main bike is rigid, and I think the suspension here is a nice change of pace on occasion. I could lose about 1.5 lbs by changing to lighter tires, but these Kenda Kross run quite smooth and have a nice feel to them.
So for now, I plan to upgrade the brakes to Avid Single Digit 3 (replacing the cheap C-Star kit on here), new brake levers of Avid FR-7 speed dial, and possibly the lower-geared crankset, and a set of decent quality Vuelta wheels.
You could probably lose a pound just by going to an aluminum kickstand. But this bike's never going to be "light" anyway. Changing the gearing sounds like a worthy goal, though you may be able to achieve the same thing by just changing the rear cluster which would be easier/cheaper.