I'm using Shimano Tourney TX right now and would like to upgrade. I can only shop online and can't visit physical bike shops. I'm leaning towards SORA R3000 but I'm not sure if it has the same BB size as Tourney.
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.
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.
If you can identify the specific part number on the crank (or just by looking at pictures online), Shimano provides a great deal of information online about compatibility.
For example, if you are using the current Tourney FC-TX801 crank, it uses a standard threaded square-taper bottom bracket, so any square taper crank will work. The SORA R3000 crankset you are looking at uses a Hollowtech II type bottom bracket, which is also threaded, but is not square taper, so in order to switch to SORA, you will have to replace the bottom bracket as well as the crank. Bottom bracket bearings wear out, so it is common to replace both at the same time anyway.
In addition to the bottom bracket, however, there are other factors you need to consider. First, SORA is a road bike group, while Tourney is for mountain bikes and hybrids. That means your bike may have a 73 mm bottom bracket (this is measured along the length of the cylinder that holds the bottom bracket). The SORA R3000 is only designed for bottom brackets that are 68-70 mm. Since the Tourney is compatible with 68-73 mm bottom brackets, it is possible that your bike has a 68 mm bottom bracket width and can use the SORA, but you need to check that.
I am basing this answer on these Shimano pages:
Mixing mountain (Tourney TX) and road (Sora R3000) components is tricky at best. But here the answer is easy because while the TX uses a square taper BB, the Sora crankset requires an outboard bottom bracket (cups threaded or pressed into the bikes BB shell with bearings that sit outside the shell). See Shimano's Compatibility Chart. These are completely different systems and you'd have to replace your sq taper BB with the outboard type to use Sora.
Further consideration must be given to the front derailleur. Would it be large enough to handle the larger Sora crankset? And would differing chainlines come into play? Furthermore, as Sora is a 9 speed system, for best shifting and noise prevention, a 9 speed chain would be best to use. A Tourney system is most likely going to have a wider 8speed chain. One can sucessfully use different speed class system of front drive components if there's only one generation difference (replacing a crankset marketed for 8 speed with a 9 speed crank, a 10s crank can be used in a 9s, 10s, 11s system, etc).
To summerize, for your switch to a Sora crankset from the current Tourney, you will need to replace the bottom bracket, chain, and perhaps the front derailleur. Then it must yield a workable chainline for best performance.