How does one upgrade a bicycle? I am a total bike novice, I can ride them but have never done much as far as maintenance. Recently, I got my old bike out of storage, it is a 1991 Trek Multitrack 700, and I want to know if I can "restomod" the bike. 

The chain, chainrings, cassette, and crankset are all fairly rusty but the derailleurs are both fine. Aside from that both of the wheels need to be replaced.

Do I need to replace the chainrings, cassette, and crankset with contemporary parts? If I install new components, how would I know that they'll fit? Would contemporary parts not get along with a new wheel?

And if I wanted to install a new gear change mechanism, would I need to get new derailleurs?
Cranks and chainrings are made of aluminium, which does not rust. You'll probably be fine with cleaning them and new cassette and chain. This isn't that expensive any more. From quick googling, your bike uses 7-speed cassette, which are still available.

Upgrading to modern parts would require replacing all transmission components, shifters, derailleurs and the rear wheel. This will be expensive and probably not worth the trouble.