(All ticket prices were recently substantially changed. They'll be updated ASAP. Don't pay attention to the numbers in the meantime).
About half of all the Kremlin territory today is still functioning government buildings, among which is the working office of the President. The other half is managed by the 'Museums of the Kremlin' and provides several separate things for you to see, each of which requires its own separate ticket.
Detailed tips about buying those tickets you can find here. This article just describes what are the options to help you decide which parts would you want to visit and which tickets to buy.
There are 4 (!) separately sold tickets for the Kremlin that lead you to 4 different places:
1) Kremlin's territory and Cathedral Square
2) Ivan the Great Bell Tower (available only in summer months)
3) The Armory Camber Museum
4) The Diamond Fund Museum
The general advice here would be to try as much as possible to see all of them. After all, the Kremlin is the most important tourist sight in Moscow. Why not take the whole package?
But clearly, not all 4 are equally exciting or important. So let's go through all of them one by one, see what they have to offer, and which ones you should prioritize if you want to select just a couple out of 4.
The list goes from most to least skippable.