Oh. LOL. I should have figured that as a novice, you didn't mean rush as the popular slang for rushing an opponent.
Anyway - you want to rushbuy things that give you a lasting advantage that outweighs the cost of whatever you are sacrificing to rush (for the sake of argument, we'll assume population (whipping) or trees (chopping) and ignore cash rushing as that comes into play fairly late, usually).
Population is usually a pretty easy call in the early game. Most capitals have food surpluses, often big ones if they have lots of seafood. So they grow fast, often right up to your happiness or health cap. Or they'll grow to the point where they're working unimproved tiles. In either case, whipping away the excess population (at least 2 at a time, to keep one step ahead of the
face that comes with each whip) doesn't cost you anything except population that wasn't going to work anyway, or that was working an unproductive tile. So - need an army to kill off a rival? Whip Barracks and some Axemen. You just conquered your rival and your science rate is at 20%? Whip Libraries (and run Scientists). And so on. Best to whip once you have a Granary (or whip the Granary first), so your population will grow back faster.
Chopping is harder to rationalize, as trees don't grow back as fast as population (or sometimes, at all). So I generally chop trees only for a key wonder (like the Pyramids if I will be running a specialist economy, or the Great Lighthouse if I am coastal, Great Library for my GP farm, etc) or if I will be axe-rushing someone and need an army faster than building/whipping it. The exception, of course, are tiles you would chop and improve anyway, like riverside grassland or forested hills - I'll chop those just to get my Granary up or whatever, since they are superfluous. I try not to chop down non-riverside tundra or plains forest, but sometimes it can't be helped... it's a judgment call. And remember that trees give your city health - don't chop down trees unless you have health to spare or are prepared to deal with the unhealth.
An extra capital city site is the AIs capital that you captured with your Axe/Chariot/Quecha/Praetorian/whatever rush. While it's not quite the same as having 2 capitals (the Palace gives the biggest boost in the early game), you'll notice that most capital city sites have excellent resources. Consequently, an AI capital will be a MUCH better city to have than your 2nd or 3rd city site, usually, and will give you a big edge in your game.