I recommend learning proper touch typing with the 'correct' finger position. Its benefit is that there's proper movement for every finger for every key, which is mostly symmetric and very regular, and it gets ingrained in muscle memory. Hence typing anything is quick even if it's not fast overall. The fingers 'know' what they must do for any letter, and it's often faster to let them do their thing than to e.g. reach for the mouse. You can actually feel when you press a wrong key.

In case of a mistake, it's faster to delete a few words and retype them than to move the cursor back and forth. And if you get an app like Alfred (for Mac) or Keypirinha (for Windows), you can invoke it with a shortcut and type a few letters faster than you open an app or click on some widget with the mouse — it's often quicker to switch apps via Alfred than via cmd-tab. Likewise, when you use a keyboard-heavy app like Emacs, calling its commands becomes a breeze.

I'd like to use Dvorak or at least Colemak, but I'm concerned that all shortcuts would be messed up. Especially since I'm a user of Vim-style movements and commands.

Also, TIPP10 app is pretty nice for learning. No fluff, no extra stuff. Initially took me about a month to learn with half an hour or so a day, and about a year to fully get used to it and develop the speed.