If you have a helper, use your multimeter to find the power drop. In all tests, make sure the probes have a good, clean surface to get a reading from.What should I check and how would I check it? I do have a multimeter so I could, with instruction, use that to check things.
First step is to put your DMM probes in the dead center of the battery posts, not on the terminal clamps. With the engine attempting to start, the battery voltage should stay above 10V. If it drops sharply when the key is turned, the battery is bad.
Next check the battery terminals at the battery posts, even if you think they're OK. Put one probe of the DMM in the center of the positive battery post, and the other on the positive battery terminal. I know it seems like you're essentially putting the meter on the same spot but you're not.
Have a buddy crank over the engine. There should be *NO* voltage on the DMM at this point. If you do (more than a tenth of a volt), then the connection between the battery post and the battery terminal clamp is bad, and you're losing your starting current there. Remove the battery terminal and sand down the battery post and the inside of the battery terminal and retry. Test the negative terminal in the same manner.
Next. Put the one probe of the meter on the negative terminal of the battery, and the other on a good ground on the engine block and crank it over. Again, there should be no voltage. If you see voltage appear in this test, the negative battery cable or it's connection at the block is bad. Remove from the block and sand or scrape clean it's connection and repeat. If the same result happens, replace the battery cable. Do the same for the positive battery cable.