How can I know if a crankset is compatible with my bike without physically seeing it?
In general, you may be able to check the bike manufacturer’s page for your bicycle. They will usually list all the components the bike is built with. However, the list may not be comprehensive, especially if the manufacturer mainly sells lower end brands.
Bottom bracket standards can be confusing, because there are a number of variations. In this case, you could have looked at Shimano’s web pages for the Tourney and Sora groupsets.
The Tourney crank takes a square taper bottom bracket, where there’s a square spindle in the BB. The Sora crank takes a Hollowtech II bottom bracket; here, the spindle is integrated with the crank, not the BB, and the latter is a set of cups with bearings. So, this is a different type of BB.
The term size is ambiguous here. However, your frame most likely has a bottom bracket shell that is designed for threaded BBs, most likely with English threads. In general, in shopping for BBs, you need to consider the frame’s BB shell type, and the crank spindle type.
Do be aware that it is usually not economical to upgrade a lower-end bike piecemeal. Bike manufacturers buy components in bulk from the group set manufacturers, and they get big discounts. It is often more economical to just sell a lower-end bike and get a higher-end one. If the issue is that the chainrings are worn out, then just in case you aren’t aware, you should be able to get single replacement rings. Also, I’m not sure the Sora crankset will be a major upgrade from Tourney. That said, if you are aware of this and you still want to upgrade the crankset alone, you shouldn’t let an Internet forum stop you.