You are looking in the right area. Most likely the diode has failed and may have damaged the magnetron. As for opening up the unit, remove it from the wall and start pulling screws. Pay attension to which ones go where as there are different types. Once the cover is removed, the high voltage section is on the right. Discharge the capacitor using a screwdriver or pliers with insulated handles before touching anything. With the cover removed, there should be a tech sheet in a pocket on top of the cavity which has component specifications.

I am not the person to suggest a meter as all mine are Flukes and not what most people would consider reasonable, but you really only need to do some basic checks so most meters should give you what you need. The diode check is only going to work on more expensive meters, but if you are hearing a buzzing sound, you may actually find the diode has overheated and possibly cracked. If it is, no need to check it with a meter. The magnetron and capacitor are basic checks too.

They are rarely called diodes (AP4352540) in the parts catalogs, but more often rectifiers, or simply HV wire.