All I ever run (and have ever run) are homebrewed sandbox campaigns. I have run starter adventures a handful of times, and I have run like, two modules in my life. 
Running sandboxes just feels natural. After the starter adventures, what’s next? What’s in the town over there? Another dungeon? Something to steal? Evil cultists?
But I realize not everybody’s like me. So here’s how I do it:
- Gather some players and decide on a system. I usually pick the system first and then seek players, but you do it how you like.
- Pick a theme. I keep it vague- romantic high medieval, gritty sword and sandal, gonzo post-apocalypse, dark fantasy, whatever.
- Write up five or so cultures. I am a big history nerd so I usually just think about the last handful of cultures I was reading about. If I had to do it now I’d probably go with Greco-Bactrian colonists, Babylonian city states, Mongolian steppe confederations, Russian merchant-princedoms, and, why not, let’s do a Malian Empire.
- Then I decide vaguely where they are next to each other. Let’s put the Malian Empire in the South, the Russians in the West, the steppe nomads in the East. Babylonians go North, and the G-Bs are in the Northeast. Why not. 
- Then the boundaries and features of the region. There’s a big sea in the Southeast, and an ocean to the North. There’s a vast desert in the south, and sparse shrubland in the east (where the horse lords are, naturally). The West is hilly and forested. There are a lot of mountains in the Northeast. The North has lots of rivers and deltas and floodlands and swamps.
- Then decide on how the cultures relate to one another. The M. nomads raid the R. merchant-princedoms regularly. The M. Empire and the B. City States have a Silk Road deal going on, with the Rs always trying to move in on it. The G.B.s constantly clash with the M nomads, whose lands they are encroaching on. The G.Bs and the Bs have constant border skirmishes, usually instigated by the colonists.
- Write down what the people in that culture are like. Keep it short. The Imperials are clean, tall, dark-skinned, and wear keffiyehs and robes. The Merchant Princes tend to be stocky and bearded, fond of drink, prone to riots. The steppe nomads wear furry hats, have facial hair, are excellent hunters and trappers, and tend to be athletic. Short and sweet. 
- Then, pick somewhere and decide on an inciting incident, something to get the party together and invested. I usually pick what I think is the most interesting area, personally. Think of a handful of nearby settlements, and write two or three words about them. “Militaristic port city” is a good one. “Secretive Forest Druids” is also good. Keep it short and sweet. You can make it up later, if it ever matters (and it might not!)
- Let’s say that I start the game by having the players try to survive a raid. I’ll tell them “The game starts in the trading town of Soralsk in the summer, so make sure your characters have a reason to be there. Soralsk is a bustling seaport by the steppe, known for its timber.”
- Watch what your players come up with, and use some of those ideas. Your players will tell you what aspects of your shared world they’re interested in exploring, so take the hint! 
- For the starting area, think of three organizations that are in ‘dynamic tension’ with one another. For Soralsk, let’s do Guildsfolk versus the Nobility versus, eh, let’s do Shamans. Currently, the Guildfolk have the upper hand, but that can (and should) change as the campaign goes on. 
- Off you go! Start the session- the players happen to be in the wrong place at the wrong time. The raid proceeds, and the players react.
The reactions of the players is extremely important. Always have the players’ actions affect the world. For every action they undertake, there should be a reaction. Somebody shows up to stop them, or to help them, or to warn them, or to ask them for help. The city they save starts to expand, and in the process, defiles a sacred river and now the river spirit is pissed off and causing problems. If the city is devastated? The refugees flee the city and the raiders continue onwards, greedy for more. And so on and so forth.
But that’s really more about running a sandbox, which I find to be fairly easy if you follow the steps above.
 Don’t get me wrong, I’ll read modules, but I am a tinkerer at heart and I am only reading them to steal their ideas and pass them off as my own.
 I don’t draw a map unless my players ask for one, or if they buy one from somebody somewhere. Sometimes they’ll make one themselves, which can be fun. Either way, distance to and from landmarks is more important than a map.
 I find that a physical trait, an interesting bit of clothing, and one cultural attitude is enough to get the juices flowing. Short people wearing floppy hats who enjoy dancing? That’ll work. Moustache-wearing people who wear colored tights and are famous sailors? One can work with that.
 If your player is playing a cleric, let them tell you about their pantheon. If they’re an elf, let them tell you a little bit about their homeland. I always use the old improv trope where you can build on what others have said but never negate. “Most elves are from tree-cities but there are also sea elves with gills and fins on their heads and I am one.” “Actually most dwarves aren’t colorblind, it’s just the royal family and their many, many cousin branches.” etc etc
 Three factions is a nice, stable situation and you can always zoom in or out if you need more or less. All the factions of the city in the example fight and squabble, but they close ranks against their rival cities. Or the Nobles come from the same lineage and are all cousins, so they work together quite well against the merchant guilds in the cities they share. And so on.