This is a big topic!
As for maps, I do all my maps by hand. If I have maps in the campaign, I tend to start at a large scale (thousands of kilometers, though I don't do scale measurements in general) to map the starting nation and its neighbors. Then I'll do a second map at a much smaller scale, maybe 100 k across, to show details of the region of the campaign's start. Then, as the campaign continues, I'll add maps to other areas as necessary.
But I don't have one set way of doing world-building. For fantasy campaigns, I have sometimes started from a map, and that's fun. But I've also done sessions with the players where we make up details of the world as we're doing character generation. Sometimes a map isn't even necessary, even in a fantasy campaign (obviously it wouldn't be necessary or even plausible in many SF campaigns). But it really varies; sometimes I have an idea and I improv most of the campaign from there, sometimes I do incredible levels of detail before we ever begin play.
Sometimes I start with illustrations of monsters and characters, who then become recurring NPCs to hang stories on, and the ideas of the campaign develop from there.
Sometimes I'll have an idea for a particular encounter or character interaction that I think would be particularly fun. Then I backtrack from there, considering: how could I get the characters there, and what sort of world would I need to build to set it in so that it would work? I built one fantasy campaign about the concept that the characters were in magical stasis for thousands of years, and then are awakened in a world that they don't know at all. Then I had to consider: why were they put in stasis? How were they put into stasis? What is the nature of the world in which they were put into stasis? So I did a bunch of world building for that past world, to help characters with their back stories. Then I worked on history: what happened to all those nations that I designed in the intervening time? What are the dynamics between the nations? What's happening in the world that causes them to come out of stasis when they do? What are the details of the area in which they find themselves when they come out of stasis?
Sometimes I have an idea for a type of game (disaster survival), and I'll just build from there. For example, I recently designed a Dread game. I was struck that Dread should work equally well for any kind of high-tension story, not just horror. So I knew the system and that it WASN'T horror; that was my starting point. I thought about "Blake's 7," and how in the early episodes there was a recurring theme of limited time, high risk, as people attempted to escape captivity in space. So I thought of a space station. Then I started building the setting, and to do that, I needed to set it in a time period (near future, mildly cyberpunk, real world, early stages of interplanetary exploration). I came up with a lot of details about the construction of the space station, and the world in which something like that could be built. Then I wondered "what is the conflict or tension?" (Dread inherently needs a threat to the characters to work) I decided that the space station is failing, which will provoke the characters to action: they must escape. Then I have to come up with reasons WHY the space station is failing. What's the reason for it? Once I had the reason, I could plan different events that could happen to the players throughout the session: set pieces that would add tension and/or prompt the players into action, where they might have to pull a brick.
Sometimes, after the initial premise, I work on developing the world through creative writing. We'd been playing in a heroic space opera game that'd been going for more than 10 years, and it was coming to be my turn to take over as GM. My initial premise was to turn what they knew of the setting upside-down: a million years in the future, when the characters they'd been playing in the current game had done something so extreme that legend still persisted of their deeds...but as a cautionary tale, rather than as an example to emulate. To help with this, I wrote various articles that the characters could find in searching the net, which illustrated the time difference and attitude change. I did a scholarly analysis of the history of one of the worlds the players knew well, detailing the social upheaval and eventual destruction of the planet. I did propaganda pieces for the current galactic empire (which was a nascent group in the current game). I did nursery rhymes that had developed around skewed, distorted impressions of the boogeymen that the current game's PCs were. In the process of writing these, I was also developing my picture of this future universe and its dynamics. Then, having developed this future empire (and its attitudes about the past), I designed the conflicts/tensions that would be in play as the game began: the things that would motivate any NPCs the characters would encounter. Then I designed ideas and options for the PCs and then we were ready to begin.
