I use a keyboard. Left hand on the number keys (the ones in the QWERTY set-up, not a separate number pad), right hand on the arrow keys. I don't think this is necessarily ideal -- on larger puzzle sizes my left hand placement can get off by a key really easily, which has blown up some times that might've been really good -- but it's what's most comfortable to me and seems to work well enough.

Honestly though I'm not sure I have great advice for speeding up. The only specific skill I know I've intentionally practiced for speed is the roll through L-shaped cages.

I do make a specific point of doing smaller daily puzzles to warm up, of doing both the warm-up puzzles and actual timed attempts in one continuous block, and of listening to music while I do all of those, but I have no idea if those actually help me, much less someone else.

So what I mean by rolling those cages is that with the way I have my hands set, there's a very quick alternation between Left-Right-Left-Right-Left in order to fill an L shaped cage. Specifically, ones with a repeated digit feels sort of like doing a drum roll rhythmically to me, so I refer to going through those cages as "rolling" them.

Now that I'm thinking about it I should probably start practicing the two common 4x4 arrangements for all 16 digits (split into 4 2x2's, or the one where the next row is just offset from the prior one by one spot), because that's probably low-hanging fruit for me.