Buy Minecraft and then?

Can I if minecraft on the official site bought the key then on any pc enter and minecraft download?

As far as I know you buy it and have the download of the Java version on your account and do not need a key, the key can be for the Microsoft Store version.

If you buy MC, you can log in to any PC at Mojang and from there download the launcher to play. All you need is your credentials to play

Since you say Key:

DO NOT BUY THE JAVA EDITION THE BEDROCK EDITION!

Then you can simply log in to the Minecraft Launcher with your account and play.