The easiest component to test is the switch, but the most likely to fail is the capacitor, which is that thing on the right with the purple wires. Your motor looks like a simple induction type. Do a web search on "why does my motor have a capacitor" and you'll know as much as I do.
Here are some tests you can try.
Spin the shaft by hand (You've already done this, haven't you?) If the bearings are seized you can probably get it to run by cleaning and lubricating but my experience is that cheap motors never last long after this.
Turn the fan on and spin the shaft by hand. If the motor starts then the capacitor is bad. Replacing this part is dead simple but finding the replacement is an exercise in industrial engineering. It's a great learning experience, especially if you want to continue fixing stuff.
For the following test you'll have to cut some wires.
Isolate the switch and test it for continuity with your ohmmeter. This is great fun with a multi-speed switch because you have to simultaneously find out how it works, and why it doesn't.
Isolate the capacitor and test it with your ohmmeter. The capacitor should first show zero resistance, then quickly go to infinite resistance. Swap the probes to see it again. If it consistently shows zero or infinite then it has failed.
As long as you have it all apart anyway, test the motor wires for continuity. It is very rare for a motor winding to break but you paid for that meter so you might as well get some use out of it.
I think everyone will agree that cheap appliances are not worth the time to fix, unless you enjoy the work and the learning. Personally I have been pulling machinery to pieces for seventy years and I never get tired of it. Sometimes it even works after I fix it.