Expand fast. There's a couple of articles, I think, in the War Academy on this. Basically, build as many cities as you can as quickly as you can. It may help to play a civ that has a strong early UU (like Persia) Don't be afraid to slap around other civs early in the game, if not for any specific tactical improvement, then only to impede their expansion. The AI seems to have great difficulty getting over early setbacks.
Don't worry, you'll get the hang of regent soon enough. The last regent game I played, I had tanks + mech inf long before anyone else (great fun), I was half an age ahead in tech and almost every civ was paying me a couple hundred gpt for various reasons etc etc.
Now monarch, otoh ....