I'm just beginning with bees, and reading a great deal.
You don't have to buy new queens.
If the bees are going to raise their own queen, they will need drones bees flying to mate the new queen. They'll need eggs or young larvae to feed up into a queen. They'll need enough honey and pollen and enough bees to look after the brood. In a cold climate they'll need enough bees to cluster and keep the colony and the brood warm enough.
Up here in Nth NSW the bees are making swarm cells. I did some splits by putting swarm cells into every split along with brood, honey, pollen.
You can put the weaker part of the split in the position of the old hive, and move the old hive a short distance away, like a meter or two. The foraging bees will return to the original location.
I did a split by taking the queen and two frames of brood and empty cells, and two frames of honey, pollen and empty cells, in a new box with empty frames at the original location.
I moved the original hive, with the brood and honey super, to a spare place in the yard. The original hive had made queen cups with eggs in them before I did this. This kind of split simulates a swarm.
Good luck. There are plenty of ways to do it. Look up Michael Bush's website.