I move a unit with a move trigger to a battlefield. I am attacker, he wants to use an action ability, but I want to use an action spell ability first. Who can use action speed spell first?

The attacker gets to act first.

Here's the sequence:

1. Move trigger resolves first. Your unit's move trigger (e.g., "When I move...") goes on the chain and resolves. During this chain, only Reactions can be played. No one can use an Action speed spell or ability yet.

2. Showdown begins. Once the move trigger resolves and the chain is empty, the showdown starts immediately (FAQ #4645). You are the attacker — you applied Contested by moving your unit there first.

3. You gain Focus. Per the rules reference (Rule 345) and FAQ #7542: "The attacker gains Focus at the start of a showdown... allowing them to play an Action or Reaction first." This is true regardless of whose turn it is.

4. You play first. You can now play an Action spell or activate an Action ability. Your opponent cannot play their Action ability until after your spell resolves and you both pass focus back-to-back in the alternating Showdown sequence.

Ruling: You, as the attacker, can play your Action spell first. Your opponent must wait until after your spell resolves and Focus passes to them.