If your begonias are the tuberous rooted ones and in pots allow them to die back and dry out. Then you can keep them just as they are in a dry frost free and mouse free place. Some people prefer to tip them out of the pots and store them in dry sand. if they are in the ground dig them up and allow the compost to dry off, then store as above.

I'm no rose expert but someone will tell you what to do.

Once your begonias have died back check the tuber / corm over for vine weevils, they love slowly munching on over wintering begonias to see them through until Spring

Roses, now is perfect time to take hardwood cuttings from roses, and also good time to move them

You could combine both activities by removing a lot of the top growth from the roses [for cuttings] prior to moving

The top growth removal will help prevent any wind rock [top growth acting like a sail] and also means the plant can get on making new roots without struggling to feed top growth