When using my action to move my major image, what is the movement speed of the image for that turn?

Major image does not have a speed at all, but it can be moved up to 240 feet in one turn under certain circumstances

As long as you are within range of the illusion, you can use your action to cause the image to move to any other spot within range.

This does not give the image a speed like a creature has, this just states that you are able to move the image around and make it look like it is moving. Nothing in the spell gives the image a speed.

Calculating an effective "speed"

However, we can calculate how far it can "move" (really be made to look like it is moving by the caster) in a turn and come by an approximate "speed" based on that. Since the range of the spell is 120 feet, the maximum it could move would be 240 feet. This would be the case if you move it from 120 feet directly in front of you to 120 feet directly behind you for a total of 240 feet for the turn. All the while you would be able to animate the illusion however you wished (walking, running, flying, burrowing, etc.).

However, this maximum is only achieved when the illusion starts at the limit of the spell's range and is moved to the opposite limit of the spell's range. This does not mean that you can move the image 240 feet every turn. If the image started 50 feet away from you, you would only be able to move it a maximum of 120 feet away from you in the opposite direction for a total of 170 feet for that turn, for example.

0 Movement speed, but can be moved up to 240'

Major Image (Basic Rules, 258) states:

As long as you are within range of the illusion, you can use your action to cause the image to move to any other spot within range.

Range on Major Image is 120', so you the maximum movement will be within that range, or 240' at it's maximum depending on where the image begins within that range (If you start it at 120' behind you and want to move it to 120' in front of you.)

But it has no movement speed at all

The image doesn't have speed or movement itself. It is only that you are moving it within the spell's range.

As fast as you want, as long as it stays within range.

It can move to anywhere in a 120' radius, within the space of a single action.

You can "animate" its movement to look like it's walking / otherwise moving naturally, in which case you probably shouldn't exceed the walking speed of whatever creature you're imitating. But if you don't care about the plausibility of the illusion, you can just have it reappear anywhere in the spell's range.