For the most part, the fancy test equipment isn't at all necessary.
The wavemeter approach has economy on its side.
I'm a big fan of using things you have around the house. Around my house, I've got receivers that can tune the HF bands. They're more sensitive than the spectrum analyzer on my bench, and the mechanism between the input port and the output port is simple. Fire up the transmitter into a load, and tune the receiver for the fundamental (does it sound pleasant? If you put in enough attenuation on the RX can you get a pleasing tone? Once you get to that point, are there "images?" Is the carrier suppression adequate? Splatter? (If you don't hear splatter when the RX front end attenuator is off, then you spent way more on your receiver than I could afford
The point here is that not every spur you hear in the RX came that way out of the transmitter. When overloaded by very strong signals, most receivers will generate images, birdies, and all manner of other things.)
Next tune to harmonics -- 2x, 3x, as far as you can go. You'll probably hear something, but is it sufficiently far down (can the S-meter help? possibly. But remember, this ain't Rhode and Schwartz we're dealing with.) I know people who measure relative amplitude with a voltmeter on the audio out. But remember to turn off the AGC.
Good luck in your new adventure!
73...