Just about every game has some sort of glass or plexiglass protection for the monitor. Some have nice decal artwork, some have painted artwork, and some are just plain with an underlay. However, no matter the case, you want the main area that people are going to be staring at for the duration of the game to look nice and clean.
However, some times you'll find the previous owner didn't take the greatest care in the glass. Maybe there's a gawdy operator or auction sticker on it, or the glass has been target of vandalism. Here's a few tips on what you can do.
Stickers: Stickers are fairly simple to remove if you have the right stuff. I like to tear at the edges that are easy enough to grab, but ususaly you'll be left with most of the center which is now difficult to remove. At this point, I'll apply some Oops! and let that soak into the remains, and let is sit for a minute, and then carefully scrap it off with my finger nails. Any adheasive solvent should work, but be careful with using strong stuff because if some accidentally works it's way to the back and onto some artwork, you'll be sorry.
Foiled stickers are a major pain in the butt, so you'll have to be extra patient with that. I do not recomment using any type of scraper more ridged than your finger nail, because you'll end up putting permanent scratches in the glass/plexiglass.
Scratches: Often you'll find cocktail glasses are scratched on the top. There's really not much you can do about scratches in tempered glass. I've been recommended things from Rain-X to Novus 2, but haven't had much luck with using either on tempered glass (YMMV). Plexiglass, however, can have scratches removed, or at least dulled. There are a few scratch removal products on the market, with probably the most popular being Novus 2. Novus 2 is designed for plexiglass scratch removal. Be VERY careful when working with scratch removal products because you can overwork the scratch and end up "fogging" the area instead. Try an inconspicuous area first. Apply as recommended, but instead of one long session, do multiple short sessions, cleaning and reviewing progress in between. Continue until satisfied.
Paint: I've seen cases where people have taped off the display area and spray painted over the artwork from the front (the artwork is still intact on the backside, though. Any paint solvent will work (once again, I choose Oops!), but take caution for any back artwork again.
After you've restored the glass/plexiglass, it's time to clean it. There are some finger print proof glass cleaners on the market, so you may want to consider using those. Otherwise, any household glass cleaner will do.
As far as what to clean them off with, avoid paper towels as they leave lint particles. A dish rag may be too abrasive for plexiglass. You can buy special wipes for glass and plexiglass, but I like to use something that is readily available, disposable, and something that you probably already have plenty of: Newspaper.
Just take a sheet, crumple it up, spray some glass cleaner onto the plexiglass or glass, and clean. Now you have a use for that stack of newspapers :-).
Novus 2 (optional)
adhesive solvent (optional)
paint solvent (optional)
You can get Novus 2 and Novus 1 at your local amusement dealer, or Happs or Mazzco or whereever. I've actually been able to pick up Novus 1 at some hobby stores.
Marquees are pretty much the same as backglasses, so use the tips above. I don't know how, but occasionally you'll see paint/artwork scraped, so you'll have to touch it up. Since monitor bezels aren't usually lit up, you can get away with touching up monitor glass artwork with normal paint. Marquees, however, have a very translucent, thin layer of paint or decal. I would suggest touching up marquees with a water based paint (so you can undo your work without hurting the marquee). You can get this at your local hobby/model store.