Recommended research paths or beelines
Science beeline :
Tech writing then Philo. Build the GL if possible(depends of difficulty) then NC. Add 1 to 3 cities before NC and build minimal military to finish the NC as fast as possible.
This beeline can serve you for many paths liek a medieval or higher eras warmonger path or a 100% peaceful game while reaching VCs like Spaceship and Diplo.
Full military beeline
Wonders to try for, given that may be VC dependent
GL, HS, HG, PT, ND, MP(if mountain on border within 2 tiles) and CI are highly recommended. For cultural victories you might prioritize Sistine and Louvre as well.
How extensively should one expand? In Civ3, REXing was a must; in Civ4, expanding too fast could crash your economy
There is no corruption. What limits your empire is your total happiness. Between -1 and -9 you suffer from growth but not production. Below -9 you suffer both growth and production(not recommended for long period of times).
There is some strategies that let you expand until -8/-9 happiness and let you gain this happiness over 0 in some turns. If you plan to build courthouses, ally a cs or trade with an AI rapidly to overcome this malus.
Usually, you stay with 1 city until NC then expand more or you can immediately expand then build libraries everywhere before NC. There is more strategies that let you expand infinitely(ICS) but don't build the NC or any other national wonders. This last approach is almost exclusively for domination games.
Annex or puppet? I've been taking the annex approach out of habit, but I might be missing something
It's usually better to annex after finishing a national wonder. It's possible to annex and rush buy a courthouse immediately to regain happiness loss. Usually, you annex when you have enough gold and happiness(and maybe after finishing a nat. wonder). Annexing a city will also raise your culture needed for future policies.
Also, how do I know the cost in gold for considering purchasing a building or unit? I've seen the purchase button, but can't seem to find the price. If I'm debating between buying more land or finishing a city's production, I'd like to know the prices.
Go into the production menu. There is a button to the left of the production icon that let you buy stuff. Scroll up and down to see how much they cost.
For all sort of questions you can also use the search function.
