Here are a few little nuggets of advice that I have learned over time:
Chop, chop, chop -- Get rid of most of your forests as quickly as possible. This provides 2 advantages. First, you get the hammer bonus that will speed along production in the adjoining city. Second, forests are a great place for your opponents to fortify a unit or two and attempt to choke your city. Don't give them this chance! To use this strategy, you will need to get bronze working, which also requires mining if your civilization does not start with it. On the flip side, if you have a few extra units exploring and happen to get near a rival city, by all means, find a forest+hill square and happily fortify on it.
Balance your defensive forces -- You can't just rely on archers and expect your city to be safe. A small stack of horse archers or swordsmen will tear right through them. It's vital to build some axes and spears to keep your cities safe.
Improve your land -- You'll need at least a 1:1 ratio of workers to cities, but I prefer to go 1.5:1 or 2:1 (or even more, depending on the situation) to get the job done right. If you don't hook up your resources, mine hills, build farms and cottages appropriately, your production, research, and crop yield will suffer and your opponents will easily surpass you.
Don't provoke your neighbor if you can't back it up -- So what if there is an unguarded worker near you. So what if you think you can pillage a few improvements. If you are not prepared for a counterattack, it's not worth the short term gain. Similarly, if you decide you have a large enough army to attack a neighbor, if you don't eliminate him (or her) from the game and lose your stack, you'd better build up some defenses!