"#041 Spiral
Give Spiral a double power action, remove all of her Gateway markers from the map and place 2 Gateway markers on the map. Place one in an adjacent square and the other may be placed in any square on the map.
Starting with your next turn, give a friendly character occupying the square of a Gateway marker a free action and place it in the square of the other Gateway marker."
Does this mean in order to move through the gateway, a character must begin the turn on one marker, i.e. only one character per turn is going to be able to move through the gateway?

No, characters do not need to start the turn on the marker. If you wanted to get multiple characters through, you could if you can:
a) get a figure onto the marker
b) get that figure off the other marker to allow the next character to "go through" the Gateway.

Sidestep is wonderful for this, as the figure could Sidestep/Move, free action Gateway, Move/Sidestep.
Telekinesis to place a character on the Gateway marker would work just as well.

Its a free action to move through, so sidestep, TK, certain other free move actions will help.

Would a character given a normal move action, not starting on a gateway marker, be able to move onto a gateway to the other gateway and continue moving from there?

No. You can't take an action in the middle of another action unless an effect specifically states you can.

The character would have to completely resolve whatever action(s) that caused it's movement on top of the portal marker and then be given a free action to be place on the other one.

Thanks much. I understand now.