Well we moved because I was PG and was about to pop, and our old place wasn't big enough. So DD at 21 months actually had a new house, a new room, a new toddler bed (crib had to go to new baby), and a new baby brother, all at the same time.
I started with putting her into our bed just for play. I would take her into our bed when she first woke in the morning to cuddle under the covers in our PJs, or randomly at different times during the day, we cuddled and read books in mommy's bed. Then, we made a BIG deal about her getting a big girl's bed. We went shopping together and she picked out her bedding. She was so excited that when the bed got here, we had all the pieces layed out in her room, since it wasn't assembled yet. She couldn't wait for us to put it together. She watched while daddy assembled her bed, and as soon as it was ready, we made her bed, cuddled and read books on her bed. That night, did the same bedtime routine as usual.
As for your falling asleep situation, are we on parallel universe? Mine does the same. I usually sing to her as part of her night-time wind-down ritual. So, I started to sing to her when I layed her down in her bed with lights out. Then after two songs, I wait about a minute, and I walk out before she is completely asleep. HTH.