Simple Green and the generic orange stuff one can find at any hardware store are great degreasers suitable for drivetrain cleaning.
The movable thing on top of the stem probably (un)locks the headshock mechanism.
I like Schwalbe One tires. Plenty of options, some of which are geared toward speed/racing, others (like those gatorskins) are geared toward flat protection. Be sure you get the right size. You can
probably fit 25mm tires, but check between the chainstays to see just how much space you have between the current tires' rubber and the bike frame (try to fit a 4mm allen wrench between the rubber and frame on both sides.) You might be limited to 23s due to wheel flex too. Rub is not good.
Bike cogs/sprockets don't really get visibly work before you should replace them.. It's more of a feel thing. The chain is probably going to need replacing before the other parts -- and you can measure pin-to-pin to see if it's stretched. You can get a specialty tools for doing so, but, really, your shifting performance will be poor and the chain chattery before you ever think to measure it...
You might consider replacing the brake pads as well since they tend to harden up and become relatively useless when they sit for years like that. Koolstop make some nice salmon colored ones but you'll need to also get some holders as the pads on there are all one piece. Some of these, for instance:
https://www.amazon.com/Kool-Stop-Bic.../dp/B0014BRW6E
Next up would be cables and maybe cable housings...
In the mean time, that's a great find and looks like it was preserved for the last 20 years... Great entry into road cycling!