Download and install the latest version from the download page as described at Installation. If you want to remove 1.x, check the installation page of the 1.x handbook.
Installing MuseScore 2 won't uninstall 1.x—both versions can coexist peacefully and can even be used in parallel. So this isn't really an upgrade but an installation of a new and different program.
MuseScore 2 significantly improved the typesetting quality to make scores attractive and easier to read. Improvements cover many items such as beam slope, stem height, layout of accidentals in chords and general note spacing. However, this means that sheet music made with MuseScore 1.x looks slightly different from sheet music made with 2.x.
It also means that scores saved with 2.x won't open with 1.x.
To prevent you from accidentally overwriting your 1.x scores, 2.x treats them as an import, which means:
If you did not manually adjust the layout of a 1.x score, then MuseScore uses the 2.x typesetting engine to layout the score. If you did touch the layout of the 1.x score, the individual adjustments you may have made should remain after opening it in MuseScore 2.x, but due to slight changes in the surrounding layout they may still not appear correct in context. If you wish to reset even manual adjustments to use the 2.x typesetting engine throughout, select the complete score with the shortcut Ctrl+A (Mac: Cmd+A) and reset the layout with Ctrl+R (Mac: Cmd+R).
While the sound in 2.x has been much improved, you may still prefer the sound from MuseScore 1.x. In that case, you can get the 1.x sound in 2.x by downloading the 1.3 SoundFont and add it in 2.x. You can do this in two steps: