Welll... you can, but probably not in every game you play.
You'll probably need to stick with the two lower levels where the human player gets a bonus in trading and some other minor things. Regent might work, too.
Then - and here's where it's counterintuitive - trade, sell, and give away the techs you develop ahead of the AI's - at least through the Middle Ages, maybe into the Industrial era.
"But," you say, "I want to be ahead of the AI's." If you don't trade your techs early you'll never get as far ahead as you want to be. By trading, you're letting the AI's do some of your research for you, freeing up your gold and shields to build your civilization to the point at which you can slide the research slider up and start researching new techs every 4 turns and leave the AI's in the dust.
Give it a try.
You might also learn to use the editor and setup a modded game that'll start with things as you describe.
(Brief editorial comment)
What you want to do may be fun a few times, but in the long run I think you'll discover that Civ 3 is at its best when one or more of the AI's actually gives you serious competition. Above Monarch level, human players almost always lag behind the AI's in research for much, if not all, of the game. That's one of the real challenges.
(End editorial)
Play the game the way you want and enjoy it. There's not a "right" way to play Civ 3 as long as you're having fun.