Welcome to CFC, laughingvulcan!
It's best to keep an open mind regarding how to use your Great People. Each leader, map, and game situation may lead to different uses. Lightbulbing techs is a favourite tactic for many, but make sure the tech is useful to your overall strategy.
For example, I'll often use my first Great Scientist to lightbulb Philosophy. Besides founding Taoism, which is nice to have in your back pocket, this also takes the shine off of Philosophy for the AI, which can help you in the race for Liberalism. Now, if I find myself in a genuine race towards Liberalism, I may use another Great Scientist to (partially) lightbulb the very expensive Education tech. But if I've got a clear Liberalism lead (say, no one else has Philosophy and/or Paper), then I'll find another use for the GS (usually in my science city, for an Academy, or if that's already built, as a super specialist).
Academies, by the way, are rarely useful anywhere but the science city. It's better to merge a "spare" GS into this city than to use him for an Academy anywhere else. Few cities, until the late game, generate enough science consistently to provide anything like the benefit you get from a settled GS in a city with a Library, University, Observatory, and Oxford University. Especially if you run the Representation civic.
Great Engineers are usually best used to rush wonders, but check which tech they'll lightbulb first; it may be more strategically useful than the wonder.
Great Prophets tend to be best for techs and/or shrines, as needed. Later in the game, once all the shrines have been built, it's usually best to just settle them in your best commerce city for the extra gold.
Great Merchants can be used for trade missions if the gold will be useful for unit upgrades and/or researching at a deficit. If neither of those options are crucial to your strategy, it's best to settle them in one of your best commerce cities. The +1 food can be
very handy. Do check what they'll lightbulb, though, just in case.
I'm not overly fond of Great Artists unless I'm going for a cultural victory. I tend to use them for lightbulbing or for culture bombs. You'd have to get and settle a GA
very early on to get the same total culture points as a great work. One warning about culture bombs, however: make sure you follow up by putting some culture-producing buildings in the city with the great work, otherwise you may find yourself losing the tiles the culture bomb gained.
A warning regarding golden ages: these are rarely useful unless you have a huge, well-developed empire. A GA, remember, adds +1 commerce and +1 hammer from every tile already producing at least one. Obviously, early in the game, with a small empire, few worked tiles, and several undeveloped tiles, this is going to be of very limited use. Also remember that each successive Golden Age requires an additional Great Person, making them more expensive. As AnitaGaribaldi said, the best time for a GA is during the space race, where the added production and commerce (for science) is valuable, and you usually have a big, well-developed empire to maximize its benefits. However, if you already had two Golden Ages previously in the game, you'll need
four Great People to have a GA during the space race! How likely is
that? Not to mention the fact that Golden Ages require you to leave several Great People just sitting around doing nothing for your civ for several turns.
! for cultural victory games i have a feel for when settling/saving is better; my general rule is pre-liberalism, i settle, post-lib, i save 'em. the ones i'm saving i never pop til the very end, since sometimes the order the 3 hit legendary doesn't line up with my original estimate.
What about in ancient Rome? Throwing criminals to lions had enormous entertainment value! 
)
) in the cities needing it the most (= the one finishing last)
