How do I manage my money effectively?
A few tips I personally employ:
Build trading posts! It might seem like the measly 1-2 gold isn't worth it, but trust me that it adds up quickly. As long as my city has enough hills and forests around it, I often prefer to build trading posts on most tiles rather than mines and lumber mills. If your city isn't surrounded by enough hills and forests, though, you should stick with mines/lumber mills on them, otherwise your production (hammers) will just not be high enough.
Specialize your cities. Gold is produced per-city and many buildings provide a percentage-based bonus to gold income in the same city; so a very economical approach is to have some cities work a lot of gold-producing tiles and then build the percentage-increasing buildings in these cities first. This is especially useful when you have a city surrounded by many gold-producing tiles.
Choose smart places for new cities. You can toggle the tile yield view (default is by clicking y, and you can also toggle this from the little button next to the minimap) to get a feel of what tiles are best for money generation. In addition, remember gold and silver mines produce a lot more gold once you build a mint in the city working them, and that many resources also give additional gold when improved - you can quickly see how much extra gold they give by toggling the resource view and then hovering over the resource icons (to toggle resource view use Ctrl+r or use the same menu as mentioned before with yield view).
Remember you can change cities to produce more gold by re-allocating citizens. You can either do that automatically by choosing "focus on gold" from the citizens window in the city screen, or you can do that manually by clicking the appropriate tile slots / building slots. In particular, remember some gold-producing buildings also have "specialist slots" where you can put citizens - this also appears in the city screen.
Don't spend on things which aren't needed right now. You want a new ship to explore with? Maybe you can wait the 10 turns it takes to make it. You have the possibility of upgrading a combat unit? Maybe wait a little and invest in culture-producing buildings in the meanwhile.
Remember golden ages produce a lot of extra gold; try actively working towards these, by investing in either happiness, great people generation, or just appropriate social policies. Remember great merchants can be used differently to gain money, though - by building custom offices or conducting trade missions.
Avoid building too many roads - they cost money each turn.
Avoid fielding too many units at once - they also cost money each turn.
Some social policies can make certain money-spending activities cheaper or money-making activities more profitable. You can hover over the gold number in the top status bar to see a rough breakdown of your spending (a more detailed one is available in the economy window) and see what aspects are worth making cheaper. For example, if you find yourself spending a lot on roads, maybe it's worth taking the Trade Unions policy (or working towards Machu Pichu).
In addition to Oaks' very good answer, consider selling everything that is not nailed down and that you don't need to the AIs. Selling them luxury resources that you don't need at that moment can give you quite a lot of gold. How much they pay depends on how much they like you, so shop around a bit.