Fastest way to move pieces across?

It depends.

In Tetris, when you hold a single direction, the piece moves that direction automatically. This is known as Delayed Auto Shift (DAS).

There are different rulesets in Nullpomino which change both the DAS and the move delay, the amount of delay for a single tap. Additionally, there are multitude of checkboxes which subtly influence it's behavior (more of an issue in DAS charge games like TGM) Nullpomino allows instant tap and instant DASing, which will cause a piece to jump to either ends of the well. If your DAS delay is shorter than the tap delay multiplied by the number of taps needed to get to wall, then DAS is faster.

There's much more discussion of DAS in the Harddrop community wiki from which I have taken this GIF from and the Harddrop forums. This mechanic is somewhat central to all types of Tetris and other falling block bames.

In games with fast DAS, and 4 state rotation (such as SRS), and low gravity, the optimal strategy for placing pieces is called 2-step finesse.

The name 2-step finesse comes from the fact that no piece should take more than 2 steps to place optimally.

The finesse for each piece in Tetris is described here with diagrams: https://tetris.wiki/0G_60_Hz_SRS_Movement_Finesse

Some fan games include trainers that will count your finesse mistakes, or even warn or end the game if you make a mistake.

Further optimizations can be made, such as avoiding placements that require double tapping. These are generally slower as pressing and releasing the same key is slower than chording together two different keys.

Fan games sometimes include 180 rotates as well which help speed up placements.

In games without all these components different strategies must be employed. It's entirely dependant on the situation and mechanics of the game.

In general Avoid tapping, and charge DAS early (like during a line clear, or preserve DAS from the last piece placed).