There are two possible ways:
1. If you want to re-arrange many products, the best way is to import them with new parameters.
2. The other way is as follows:
- go to the Catalog → Categories page
- find the necessary category in the left panel, click it, then open the "Category products" tab.
- press the "Remove all" button; the products will be removed from this category, however the products themselves will not be removed.
- then go to the "Category products" tab of the destination category, press "Add products" and add all necessary products at once there.