For starters, let's look at the "small" map size, which is the default. While you can certainly win with fewer cities, if you want to play
well you're going to want to get at least 7 or so, preferably 10ish, cities by the midgame. In civ6, things are usually identified as "tall" options by giving bonuses to city with governors/maluses to those without. There are 7 no matter how big you are, hence I used 7 as a minimum. But this really is your starting point; your power base, if you will, going into the mid-late game. I'm just going to talk about the cities you settle yourself; you can also acquire them by taking the ones other people found. (Indeed, early war is very very strong!)
How tight to pack them? It depends on how you want your economy to be based. There is no longer a per city penalty* and there are upsides to having more cities vs fewer, so we will focus on what the cities do.
*Number of cities affects the era score threshold for dark and golden ages. More cities makes it harder to achieve, after a point.
Some civs really thrive based on their terrain. Often they have a unique improvement (Australia, Inca, Sumer.) Australia is a good example. Their Outback Station gets big bonuses for being near other outback stations, so you want to leave open spaces between cities to make sure you get the most out of them. This would push you towards spreading out cities a little; 4-6 tiles.
Some civs really thrive by using their districts. Japan is a good example. Getting extra yields by placing districts next to each other would suggest you might want to clump cities together so you can build a sprawling district megalopolis. As
@ArallanKing mentioned, Germany is another example of this. Because they have a special Industrial Zone interaction with Commercial Hubs, you want to place as many of those together as possible. (See my signature for efficient ways to do that.) This means you want a lot of cities packed tight, so you can fit as many of these clusters of districts in as you can.
Generally, though, Civ6 has a lot of flat yields and few % based boosts; this pushes players to have more copies of districts by having more cities. There are also regional effects, like stadiums and factories being able to give benefits to cities up to 6 tiles away- incentivizing you to place cities so that you can cover the most cities with the fewest entertainment complexes/ industrial zones. Regardless of how tight you pack, you will want a couple cities that have some breathing room so they can become large and able to push out a lot of production when you need to - building an army, getting a wonder, doing a space race, etc. You can even do both - have tightly packed clusters of cities with some space in between the different clusters.
So the best leader/victory agnostic advice I can give is "fill up the space you have as best you can." A civ like australia may place their first handful of cities farther apart (to claim more of the map) and fill in some of the holes later; a civ like germany or japan may want to settle like a slowly spreading carpet so they can develop their clusters of districts early.