Like, should I run a CE or a SE if I'm going for a spaceship victory? How about cultural? Diplomatic? Domination? Conquest?

And I just want to clarify on this

A CE is where you only build cottages and pretty much no or just a few forges/farms/mines/windmills and a SE economy is where you focus on great people and building tons of forges/farms/mines/windmills?

You should decide to run a SE or CE based on what leader you have and what land you have - either can achieve any victory goal.

A SE is where most or all of your research comes from specialist.
A CE is where most or all of your research comes from cottages.

You still build forges/mines/watermills in both (though a CE often monopolizes rives sides, so few watermills), as you need production in both. And you still build farms in both, as food is still vital for both, however, yuo tend to build more farms in an SE to feed the specialist (it has also been called a variant of a FE or farm economy).

What is best for what? Well, I find that a CE is better for the late victories, i.e. space launch (never did bother with cultural). That is just my experience, probably because my play style favours CE over SE (I almost always use a hybrid). In single player, they are about equal in warfare with the SE being a tad more powerful as it starts strong and have good wipping capabilities.
In multi player, SE rocks, as CE is harder to keep healthy with all the usual raiding.

Best to run a hybrid.

Generally, conquest and domination ar eeasier with a SE as you can your research from specialists and bulbing, allowing you push your slider to mostly gold while "acquiring" AI cities.

The CE is generally more applicable to longer games with peacful endings, such as space or cultural.

OF course you can mix and match as much as you want.

Generally, a Specialist Economy is

- less powerful from the midgame onwards
- more demanding in terms of micromanagement
+ less affected by large-scale unhappiness as using the culture slider hurts less
+ more flexible; with your high number of farms you can easily whip, work mines or run production specialists if the occasion warrants it
+ less vulnerable to pillaging

Although it doesn't feel that way, maintenance sets back a SE more because the output per population point is lower. On the plus side, you won't completely break your economy unless you're running a deficit at 100% gold; your specialists will still be doing something useful. A CE curls up and dies if you push it too far.

I wouldn't say one is better suited to a certain victory type, but Cottage Economies don't lend themselves well to constant warfare, favouring more of a 'build up, crush someone' approach. With a SE you can war just for the hell of it.

heres my run down (not an expert)

space- CE
cultural-CE in core cities, need a strong GP farm
diplo- doesn't really matter; mass media is a tech route that is very easy to pop- but all that means in 1 GP farm
domination/conquest- I suppose FE (like SE). SE economy isn't really much of an economy- the best SE's are centered around great people and the GP farm, not the actual turn by turn research of specialists

For difficulties Monarch and below, just do whatever you want and have fun.