You’re getting good answers here. Let me try to summarize and add my own comments, presuming that you don’t have all-wheel drive, which adds its own issues.
If you’re going to have tires with different tread depths (new versus used), it’s strongly recommended to have the better ones on the rear. Most tire shops will insist on this these days. This fact wasn’t understood well in the past, so a small number of people still cling to the old ways and disagree with this. With tires of different tread depths, oblivion has pointed out the drawbacks above.
It’s recommended to have the same tire and reasonably close tread depth on each axle.
It’s preferred to have the same tire on the front and rear. If you can’t, then try to use tires with a similar tread design.
It’s ideal to have the same tire with the same tread depth all around.
All of these rules are for safety reasons, mostly to keep the car pointed in the desired direction on a wet or snowy road, especially when you’re in a turn.