Multiple river tiles is key if possible (hydroplant is worth giving up settling on a hill for easily)
Also, flat freshwater tiles (from lakes or oases) are ok too.
Think about where your expansions are going to go and count at least 5-6, ideally 7 tiles away. This way cities don't have to clash about who gets to work which tile.
Coast is good only when there are several sea luxes or resources like fish/atoll; if it's clear you can't create a 2nd coastal city, don't bother.
Look for pantheon "theming"; quite often the map generator will pair wine with incense, or pearls with gems, or wheat with citruses, or buffalos with elephants and boars/foxes. The more "theming" tiles you have in your capitol, the more ridiculously strong your pantheon will be if you manage to get it. For example, say, normally I'd value a salt tile over a flood plain citrus, but if my location has 4 wheats, then I value the citrus more.
Look for wheat, deer, and bananas. These resources are the best early game (deer and bananas won't be that good later but riverside wheat is awesome the whole game). Avoid jungle and blank desert tiles unless your terrain is godly enough you have a good shot at Petra.
Have at least one lux within 3 tiles. It's OK to have a lux in the 4th ring because your capitol acquires tiles more quickly than other cities, but 5 is maximum; your borders will not extend 6 tiles away unless you use a GG. Multiple copies of luxes are nice for early gold and is your main income source early game, but if your city's first/2nd ring tiles are awesome (or you are Shoshone) and you don't need to buy tiles, gold is less important so sometimes I get by with only one lux copy in my capitol.
Don't be tempted by salt or marble to give up hydroplant, or otherwise a much better spot. The only thing I'd give up hydroplant for is probably Uluru or any faith wonder. (religion is about the only thing that comes close to compensating for the loss of 20+ hammers late game)
On Immortal, don't settle for bad starting terrain, explore and hopefully some AIs have better lands for you. You can forward settle them easily on immortal.
Also look for choke points and how you can block off a plot of land from enemy settlers for you yourself to settle later. I love Inca because of this.
Otherwise, it's on a case-by-base basis.