Well, pretty much very likely the lamp is blown, it almost never is the microwave circuitry that blows instead. Just note, before doing anything disconnect the microwave from the mains, and put the plug in the nice little multiuse socket holes that it came in from the factory. This is mostly for your safety, there are voltages inside the microwave that are lethal, and a capacitor that can store a lethal shock for a few minutes there as well. Do not touch any of the high voltage area at all, even with the power off, unless you have had some training in the hazards involved.
However, to test the lamp it is very simple. Disconnect the lamp ( press the 2 small tabs on the spade connectors and pull them off the terminals) so you have 2 spade connectors, almost all lamps in modern microwaves are a 15W lamp in a black bakelite base with 2 spade terminals on it, and are almost all interchangeable with the only difference being if it is a 115VAC lamp or a 230VAC lamp. To test, take your multimeter and set on 2kilohm range, and test the leads by shorting them together, which should give a reading of around 0 to 5 ohms, depending on the leads you have. Then measure the resistance between the 2 spade connectors. Any reading that is not beyond full scale ( around 1k to 100 ohms, depends on the lamp) and which is not the same as the shorted reading means the lamp is good. Otherwise new lamp (a whole $2) time.
While you are there, clean the inside of the lamp cavity, it gets very dirty, using a clean dry cloth to wipe most of the dust and grease off the inside. Then put in the new lamp, secure wth the single screw or bend the tab back over, and put the 2 spade connectors back on the terminals, close up the microwave case ( and put all the screws back please, and in the right holes as some do use different screws in places), put back in position and place a cup of water inside the microwave cavity and test it. Lamp should work, and water should get hot.
