i never played civ V but here's basic civ IV strategy ... everything below is a generalization i'm not going to mention exceptions...
build a worker first.
the first thing you research is whatever lets your worker improve whatever food resource you have (hunting if deer, agriculture if corn, etc). then tech bronze working and the wheel pottery and writing. Early religious techs are a noob trap.
after you get your first worker, grow your first city until it reaches the size it needs to be to work all good tiles -- good being anything that produces 5+ of some combination of food, production or commerce. You can build mines on rivers to make good tiles to work, but if the mine isn't on a river or a special resource its probably not worth working yet
Once you're done growing out your city, build a settler, and move your worker to forests near your cap to chop them (requires bronze working). chopping forests gives you hammers for faster settlers. Then start building workers and settlers in no particular order until you have about 4 cities and 4 workers (keep chopping forests all this time)
Next you want a source of research... connect your cities with roads for trade routes. if you have rivers build cottages (needs pottery), if you have a lot of food build libraries in your food cities and run scientist specialists (you can hurry production of libraries by switching to slavery and "whipping" them.). if you have a lot of coastal cities, build the great lighthouse (requires sailing and masonry).
you'll also need barbarian defense at some point but probably not a big issue on prince difficulty... should probably build a few military units just in case... should also switch to slavery so you can hurry production of military units in your cities in case a barbarian gets too close.
after that you need a strategy, you can build military units and take over the world. or if you have extra land you can build more settlers and workers and tech to currency to support them with trade routes (extra cities beyond about the 3rd or 4th cost a lot of money to support) or you can go for liberalism to get yourself a huge tech lead. or you can go for music and religious techs and try for a cultural victory (read some guides for this). or just build wonders.
another noob trap: don't build too many buildings, most of the buildings in this game suck. Exceptions: granaries, libraries, forges, and lighthouses. Barracks if you're building a lot of military units in that city. But settlers and workers are higher priority than buildings. You want about 1 worker per city and you want about 6 or 12 cities. (don't go past 6 until you have a good source of commerce though, or you'll go broke since they cost money to maintain).
don't build markets, banks, aqueducts, coliseums, temples, monastaries, etc...