I had this problem too when I first started playing on the Regent level. Work on micromanaging your first couple of cities so that you can get maximum growth and shield production from them. This way, you can crank out the settlers faster and start dropping cities.
My approach is to keep on dropping cities until there is no more room to expand and then do a little more micromanagement to increase shield production, and then build a formidable army. Mainly offensive units with some good defense around the borders, and in cities that need extra happiness. If you are not a comfortable size at this point, it is good to pick a civ that borders your own and concentrate on building an army to cripple them. Usually, the most powerful of your neighbors is a good target (since the other ones aren't getting any bigger, you will typically become the most powerful civ after you have crippled your biggest opponent). Switch as many of your cities as you can to build offensive units (mainly swordsmen, but archers will work too if you're in a pinch). Choose a few key cities that you are intent on taking and that will seriously cripple their empire, while boosting your own. When you have a good sized stack of units near each city, initiate a war with that civ and go for those cities. (Once the attack has begun, they will send the counter-attack at you which may make you want to go after smaller cities instead...but do not do it.) Stay with the attack. Once you've taken the key cities, either raze them and plant new cities in their place or build immediate culture to keep it from defecting back over to your enemy. At this point, you can make peace if you choose or continue with the attack. Either way, your empire will have expanded to a much more comfortable size that will give you some serious control over the remaining outcome of the game. (That is if you play your cards right.)
There are some other great articles in the War Academy and on these forums concerning some of these issues. Try searching around and reading some of those ones too.
Good luck with future Regent games!