You will get plenty of expert advice here... in time. (the initial writing started before Duke's post appeared!) For now, however, you are getting my advice...
First off, read up on 'super science cities' (SSC), because such a city gets treated much differently than other cities..
For the most part, especially in the early game, look to minimize the number of improvements you build. That said here goes. Others will correct me if I get any technical info wrong...
1. Temples (upkeep 1gold/turn): The maximum number of citizens a city can have before unhappy citizens show up is based on game level (1 at diety, 2 at emp...6 at chief). Try to hold off building temples as long as you can without having disorder.
2. Barracks (1g): If you are a warlike player, build'em. If you are a peace lover like me, don't. Even if you want barracks, try to pick out a couple of cities, put barracks in them and build as many of your units in those cities as possible. Also, if you are warlike, give thought to building Sun Tzu's wonder.
3. Harbors (1g): If you are a peace/democracy/trade player, harbors are often the first improvements you build voluntarily. By making regular sea squares produce 2 instead of 1 food, they allow you to grow cities to produce a lot of trade.
4. Libraries (1g): The SSC always gets a library. Otherwise, you have to decide whether a particular city produces enough science to be worth investing in a library. Remeber that there are two factors which determine how much you get from a library: How much trade (and therefore science) the city creates and how early it is in the game.
5. Marketplaces (1g): Increase tax and luxury by 50%. Never build a market in a city which produces less than 2g/turn at a tax rate of 10%. Otherwise, it is much a like a library: More worthwhile in big trading cities and earlier in the game. If you use celebration days to grow your cities, markets will more then pay for themselves. Another consideration is that in the advanced stages of a game where your empire is a democracy, most of your $$$ will come from trade, not taxes.
6. Granaries (1g): Grows your population faster. Usually in SSC. At high difficulty settings, almost never. This is because on the high levels, you are in a race against time to get your cities set up so that you do not start getting unhappy citizens and therefore disorders. At lower levels, look into building the pyramids, which acts as a granary in every city.
7. Aquaducts (2g): Necessary for a city to grow if it is size 8 or bigger. Always build in time to prevent your cities from having to wait to grow.
8. Colleseums (2g): Try to build the Mike's Chapel and/or Bach's cathedral Wonders to obviate the need for colleseums.
9. Banks (3g): Same effect, which is cumulative, as market places. Never build ina city which produces less than 6g in taxes at a tax rate of 10%. Otherwise, the analysis is the same as for markets.
10. University (3g): Same effect as library, is cumulative with library. Always in SSC. Same analysis as library, but remeber that the build and upkeep are greater plus universities become available later in the game.
11. City walls (0g): Only in cities which are threatened.
12. Coastal fortresses (3g): Doubles defense against warships. Once the ai's start to field navies, you will need these in important coastal cities. Not before.
13. Port facility (4g): Like a barracks, but for ships. In the mid to late game, build a couple in coastal cities with good shield production and startegic locations. Build all of your warships there.
14. Stock exchanges (5g): Same as markets. A city needs to produce at least 50 trade arrows to even consider one of these. Your SSC will definitely get one and, most likely, none of your other cities.
15. Research labs (5g): Same as library. You get them too late in the game to be worth anything. However, building SETI (research lab in each city) is well worthwhile. If you somehow get beaten to SETI, build on in the SSC, otherwise, these are a waste of money and shields.
16. Factories (5g): Build in your 2-5 best shild producing cities, but usually not the SSC. Never build a factory without a power plant and never build a powerplant. Confused? Well...
17. Power plant/hydroplant/nuclear plant/solar plant: Only effective in cities with factories. With a factory, it will double shield production. Do not build these. Build the Hoover's Dam wonder (acts as a hydroplant in all cities) instead, THEN build factories in selected cities. If you do not get Hoovers, shut the game off. Just kidding, if that happens, build the power plant/factory combination in selected cities. The only difference among these improvements is that they each one produces successively less pollution (more specifically, they allow you to produce more shields in that city w/o pollution). Hoovers will reduce pollution more than enough.
18. Cathedrals (3g) Bigger temples: Build Mikes chapel instead, it acts as a cathedral in every city.
19. Mass transit (?g): Reduces pollution from large city populations. You will need them when your cities get big.
20. Recycling Center (?g): Reduces pollution from shield production. With hoovers, these are unnecessary in all but the most mammoth shield producing cities. Might be cheaper to build an engineer to clean up...
21. Superhighways (?g): +50% trade from squares with roads in them. Always in SSC. A city probably needs to be size 15 or bigger (and with mostly land, not water squares) to benefit from superhighways. That is an estimate. A much smaller city which is active intrade might qualify. Massively improves the value of trade route to/from city since payoffs from trade deliveries (i.e. caravans and frieght) are based on trade produced in that city.
22. Off shore platform (expensive): Each water square produces plus one shield. Mostly a white elephant. It would be good for cities with mostly water squares, but becomes available too late in the game even for those.
23. Manufacturing plant (expensive): Like a second factory. Look at the cities you chose for factories. Do not build manufacturing plants there, either! These come too late in the game to be worthwhile, unless the extra shields allow you to do something very specific. (I once built one in a city with a barracks b/c it allowed me to build a (veteran) howitzer every turn!)
24. Police Stations (?g): Reduces unhappiness caused by military units in the field in Republic and Democracy. Try to get the Women's Sufferage wonder (acts as a police station in every city) instead.
25. Sewar system (4g): Needed for a city to grow beyond size 12. Always build in time to prevent your cities from having to wait to grow. Size matters!
26. Airports (?g): Allows one unit per turn to instantly travel from one city w/airport to another. Build in strategic sites. For instance, if you have a single colony on a continent held by another civ, airport will allow you to get more units there fast. Always build in your SSC, so that you frieghts can travel instantly to and from there to expedite trade. Incidently, also acts like a barracks for air units.
27. SAM missile battery (Expensive): Never unless you are having a big problem with fighter and cruise missle attacks. Even then, it is probably better to destroy the source of the attacks rather than build one of these.
28. SDI Defense (Expensive): Protects city and area a little larger than the city radius from nuclear missiles. Nuclear missile require the rocketry tech AND the Manhattan Project Wonder to be built. Build your SDI's before this happens or else

!
29. Supermarkets (?g): More food means bigger cities. Pretty much want to build these everywhere, or at least everywhere that you have a bunch of squares irrigated. Also, remember that the supermarkets do nothing by themselves. They allow you to 're-irrigate' land making it farm land. So do not build supermarkets in cities where you do not have engineers handy to do the irrigating.
I think those are the main ones. I could not recall the upkeep of every improvement and I rather suspect that I am wrong about some of the ones I did. Others will correct me. And obviously, all advice is subject to the caveat that you should experiment and see what works for you!