Australia, Indonesia, and Cree - as you might guess, I like building massive cities and playing a fairly balanced game. With any of these civs I can generally get to turn 150 or later without having decided which victory to go for (and often, can have a viable shot still at any victory path, though I often skip religion with Cree).
Australia - stupidly overpowered, but that's kind of what makes them fun. Get work ethic and a couple 5-9 adjacency holy sites, and once you research theology you're riding a production and faith economy train. The coastal city bonus lets you settle more locations (and can get you a crazy Mausoleum city), outback stations let you have productivity from flat land, and the pasture culture bombs mean you get lots of land pretty quickly. The theater square adjacency lets you keep up with culture, and once you get to a point where there are some cities/city-states that have been conquered, you can go a-liberating to get some incredible production bonuses and build all the mid-late game wonders. Or, if science is more your bag, just build high-adjacency campuses in all your high-population cities, slot in rationalism, and race to the stars.
Indonesia - the reason I am excited for the city-state picker. Indonesia with Auckland and Nan Madol on a water map (which I got organically once) is insanely fun - massive water-focused cities with high production and more or less nothing but districts on what little land you have. The faith from cities on coast give you a good shot at the first pantheon, and ends up adding up to a pretty nice faith economy throughout the game - useful when you want to buy your superpowered Jongs to explore (and conquer) the world. My favorite civ for spreading an empire all around a water map and basically just taking over the world.
Cree - one of the most fun early games in my opinion - the extra trader plus land from trade routes means you can put together a pretty substantial early empire (and if used strategically with borders can often choke off a nearby civ from expanding much). The Okihtcitaw is kind of meh but the beginning promotion means you can move a little faster from the get-go, and the increased strength helps with survivability. Two of these can get you good early exploration. The Mekewap gives you early production and housing, and probably most importantly is a major early source of gold (which can be hard to come by). They also scale up nicely to remain relevant throughout the game. The shared visibility helps continue your exploration as you get into creating alliances. Also, somewhat underrated point - I find the Cree to be one of the best civs for chaining together multiple golden ages from Classical on forward. Both the unique unit and infrastructure can be used in ancient (though I've sometimes held off on the Mekewap to get the points in Classical), the scouting bonus means you get goody huts, barb camps, and meet civs/city-states, and the early growth/production can help knock out a few early wonders if that's your style (or a variety of districts). When you hit mid-game, the shared visibility means you're often the first to meet all civs, you discover all natural wonders, find additional city-states, and can scope out a good path for circumnavigation. I've never had an issue keeping golden ages once I get into late game, but the ability to have and maintain large cities I think tends to help there too (remaining adjacency bonuses, wonders, etc.).