We moved DD into a big girl bed when we moved house. She was 17 months. We went straight to a single. We kept the mattress on the floor until she was 20 months and could climb in and out of the bed on her own.
The other advantage of having the bed on the floor to start with is it's not far to fall. We put a pillow on the floor next to the bed in case she did fall out coz the mattress we have is about 20cm high. Now that she's nearly 23 months she doesn't fall out of bed (but we still have a side rail just in case), she can climb in and out on her own and loves having a big bed.

When we moved her into the big bed, she was a bit unsettled for about 3 nights, and then after that she started sleeping through the night unless she wasn't well. She also stays quiet for longer in the mornings coz she can get up and play and will happily do that for a couple of hours before we even wake up. - great on weekends!

Edited to add: When you move, make her room the last room you pack up in the old house, and the first room you set up in the new house - this will help reduce her stress during the move and it will give her something to do while you're busy.