As it happens, I’ve just been playing Civ3 myself (vanilla, not expansion). Some tips I’ve found:
As long as you’re producing at least one beaker a turn, it’ll never take longer than 40 turns to get a tech. In the early game, it’s almost impossible to beat that, so try to stay at exactly one beaker a turn for the first two or three techs, to save money (which you can later use to buy techs from other civilizations). Set your research slider to either 20% or 10%, and if you have a city that needs a specialist for morale reasons, make the specialist a scientist and set research to 0%.
If you’re an expansionist civ, start by researching the techs which are least useful to you. When barbarian huts give you a tech, it’s never the one you’re currently researching, unless that’s literally the last tech of the ancient age. With 2-3 scouts out looking for goodie huts, and the expansionist improvement in hut outcomes, you’ll usually end up getting your valuable techs quicker by hutting than by research. Likewise, if you’ve contacted any other civs, try to get as many techs from them as you can right before popping a hut, so you don’t get one that you could just trade for.
I may be a bit unusual in this, but I put my military on dead last priority: If a city has a choice between making something peaceful and something military, I always go for the peaceful improvement. I mostly get through early wars by using my tech/economy advantage to bribe every other nation on the map to fight my enemy for me.
That said, however, temples are not a big early priority, unless you’re Religious and can get them for cheap. Under Despotism or Monarchy, you can use up to two Warriors per city to improve morale, and they’re a lot cheaper than temples. By the time you’ve got The Republic and can’t use military police any more, you’ve probably got a lot of temples built anyway, multiple luxuries, and (if needed) the cash to spare on a little entertainment spending.
Contrary to what Youngmade said, don’t look for irrigatable squares for your first cities. Under Despotism, if a tile would produce three or more of something, it’ll produce one less. So a grassland is worth two food, and an irrigated grassland is still only two. It’s only worth it to irrigate if you’ve got a cow, wheat, or floodplain (so you’re losing that one anyway), or if it’s a plains. Mostly, the best tiles you can get in Despotism will be grass-with-rock, which you can mine for 2-2. So try to find locations with bonuses or grass with rock.
Mind your early trades. If someone can’t give you fair value for a tech, then don’t sell it. They’ll have more later; you can sell it then. Also, world maps are a lot more valuable in the early game than they are later, when they get tossed in for free on top of nearly every trade: Don’t sell your world map for the first time until you can get something really good for it (but then sell it to everyone for whatever you can get, because they’ll all trade it between themselves anyway).
If you want an early wonder, it’s absolutely essential to pre-build it: As soon as you have a city with good shield output, start building a palace in it, or whatever wonder’s available, even if it’s not one you’re interested in. By the time it finishes, you’ll have had a chance to switch it to what you really want. Don’t bother with the Great Wall or the Oracle at all; get the Colossus if and only if you think you’re likely to be able to get Copernicus and Newton in that same city; and get the Great Library only if you’re basing your whole early strategy around it. The Pyramids, Lighthouse, and Hanging Gardens are usually good.
And the speed of your computer makes absolutely no difference. If you’re still using medieval tech when everyone else is industrial, that’s your fault, not your computer’s.