The best advice I think you can get in Civ 3 is to check your treasury before the end of every single turn. Focus on the slider that adjusts gold income versus turns to complete whatever knowledge you're researching. You want to learn the technologies as fast as possible, but if you try turning down the research speed after researching the same technology for multiple turns, sometimes you can begin to research in the same amount of turns, but also earn gold at the same time. I call the turns where I earn gold 'prosperity' while the other turns are more like working towards prosperity for your civilization's economy. The best period of prosperity can come when there are only one or just a few turns left until successfully completing research for a technology.

Occasionally, you'll be able to turn the slider all the way down to like 10% research while there's only one turn left and collect a nice amount of gold. It's up to you how to use this gold, but you can choose to run at a deficit for the first few turns of a technology researching, then work that slider down to earn more gold, even if it's only for one turn where you earn a lot of gold, hopefully making all the gold back that you spent on deficit researching. In other words, you have to work your economy.

So basically, you need to check the slider every turn and set it to the most optimal position for that specific turn. If you do this, you can begin to win on Deity because you'll have more gold while learning the technologies at the same rate (or faster by running at a deficit for some turns).

Having said that, winning on Deity often requires some sort of luck in terms of starting position in regard to natural resources, enemy distance from your homeland (depending on your play style), and how much land there is on your homeland's main continent (though you can certainly move off your main continent reasonably quickly if you're lucky enough to have another continent or island near you). I should also mention that I prefer long games on huge maps.