How long should I wait before I reassemble my freshly painted kitchen cabinets and drawers?

If you used latex or a water-based paint I would suggest a really thin coat of Vaseline where the doors touch the framing. Until a paint has gone to what is called Cure they will exhibit blocking (sticking to another freshly painted surface) The Vaseline will keep them from "touching" each other and won't effect the color or the curing process and will wear off about the time that the paint has cured.

Put the hinges on the framing then add the doors to them. Make sure to tighten all the screws after everything is attached, makes sure of correct alignment.

Regardless of which way you go, identifying the style you're looking for will help narrow your focus on the available choices and also provide some guidance to kitchen designers or cabinets makers you might work with. Paint is an easy, and inexpensive way to give simple cabinets a modern look. The first step is choosing a color, draw inspiration from something in the room, a rug, fabric or an accessory you love. Cabinets are chosen first during these meetings, because they are the biggest investment in the kitchen. And the kitchen is one of the biggest investments in the house.

Most paints get harder with time, I think you should be able to put the hinges and handles on. 3 days should be fine.

I would definitely put the hinges on first before you hang the doors.