I have a road bike (2013 Schwinn Tourist) with a sealed bottom bracket (107mm spacing) with a square taper and triple chainwheel (48, 38, 28) assembly that is one of the inexpensive riveted types. I want to upgrade to a crankset that has replaceable chainwheels. Can I use any crankset of my choice? If not, what are my choices?
There are a number of factors to consider here as others have stated in comments, this upgrade may cost more that you are willing to pay given the cost of the bike originally. I'm going to assume you know that and answer as if you are going to do it regardless.
This bike is an 8 speed rear cassette, which controls the type of crank sets you can use without upgrading more parts. You will need an 8 speed crankset, to make sure your chain is compatible.
The BB is a "standard" 68mm threaded shell, which can be adapted to a variety of options for cranksets.
The SHIMANO M361 Hybrid crankset 48/38/28 tooth in square taper is an inexpensive and reasonably likely direct fit replacement.
It is impossible for me to judge the length of BB spindle required after you change the crankset, however, so you should be prepared for the need to change that also.
Depending on the offset of the chain rings on your original crankset, and the difference in offset on the new one, you may need to get a longer or shorter BB spindle to adjust the chain line with the new crank.
The best way to do this is to measure from your seat tube to the center of the teeth on your middle chainring with your original crankset, and note the measurement.
Install the new crankset, and measure again. Note the difference. If the new measurement is larger, you need a BB with a spindle which is shorter by the difference in your measurements. If measurement is shorter you need a longer BB spindle.
Since this can't be checked ahead of time, you be to be prepared to have the bike non-functional while you get the new BB.
Then you need to readjust the shifting on the front derailleur to match the new crank. This may require a new from derailleur cable and housing, or it may not.
This will cost a fair bit of time and work, and between $100-150 dollars. Whether that is justifiable on this bike is up to you, but it is approximately half the cost of the bike, not including labor.
I hope that is helpful.