I definitely prefer a map for battles. Exploration is a little easier with a map too. Role play and downtime does not require maps/miniatures at all. Graphics are always nice to have though if you can.

I'm a fan of using what works for individual moments. If it's a small battle in a square room, there's little need for minis. But it helps to have them nearby if careful positioning is important. Much of the time, you can even just put them out on the table without a map underneath, just to help players get an idea of where things are at spatially. 

I'm a fan of at least drawing a map on paper for encounters. You don't need to go with big production value. And for a walk through the city market, you can just describe that. But some people are worse at visualizing than others. If you say "You're at one end of a long a rectangular room with the far wall curved into a semi-circle. Two thirds of the way up the right hand wall, there are two doors side by side. There's another door halfway up the left hand wall, and four doors leading out from the semi-circular wall at the end..." By then, you may have already lost a few people. But it might take just 5 seconds to sketch that out on paper.

If combat starts, or a trap is set up, you can just have people make marks on the paper map. Or if you have minis, you can use them then. But I'd say just use whatever you need to in order to make sure everyone understands what's going on. If you're ready to sketch things out or use minis while needed, then don't stress what you 'should' do at any given moment. Do what you need to to make it work :)