Generally, you cannot "selectively choose my layover to be the next day". At that point, you are taking two separate flights. You will have to retrieve your luggage and go through customs and immigration controls. Then the next day, you will have to re-check your baggage and proceed through the normal procedures for taking a flight, including security.
If you do have a long layover, you might be able to leave the airport (depends on whether a visa is required, sometimes you don't need one at all, other times you don't need one as long as you are "in transit" and stay in the airport), but you would have to go through immigration controls, and would have to pass through security again when you came back to the airport. So you would have to have enough time to pass through security, as well as any travel time. Bear in mind that the airport might not be all that near to the things you want to see. And do you really want to take the chance of missing your connecting flight if you get stuck in traffic somewhere?