Before writing up this section, I should mention that the previous ones were written based on the 1.21f patch of Civ3, while the following ones were written based on the final 1.29f version and the Play the World (PTW) expansion. I have gone back and made some changes where appropriate in the older sections, and on the whole what I wrote in July of 2002 holds up well for the final version of the game. Everything else that I write from this point on should be current information for the game, unless Firaxis releases another patch for Civ3.
Now we get to the actual subject at hand, the subject of the different techs available to be researched in this game. If you think I'm going to go through all 77 techs that exist in this game, think again. I'll touch on the most important ones, what they do and what new features they enable, while skipping useless ones. Since research in the earlier ages is when the game is most in the balance, I'll be focusing intently on the earlier techs and give the later ones less detail. I will also point out that you can figure out the relative cost of each tech by looking in the Civ3 Editor, which has lots more good information.
Bronze Working: Scientific civs start with this tech, which has an average cost (3) compared to the other starting techs. Obviously the major thing that this tech does is open up the ability to build spears, which you pretty much cannot play a game without using. Civs that have their unique unit (UU) based off the spearman (Greece, Zulus, Carthaginians) will probably want to get their hands on this tech as soon as possible. Also note that this tech allows you to build the Colossus. I would say that getting this tech should be a pretty high priority for any player, as your cities will be easy pray until you get spears in them. Keep in mind too that this tech is needed in order to research the very important Iron Working tech.
Masonry: This is the tech that Industrious civs start the game with. It is one of the most expensive starting techs, with a cost factor of 4. Masonry does a couple of different things, allowing you to build city walls, the palace, and the Pyramids wonder. Since it's highly unlikely that you will need to build city walls or the palace in the early game, this is not one of the most desirable early techs to get. Pick it up when you can, since it opens up the way to a lot of other good techs, but this tech has a much lower priority than many of the other starting techs.
Alphabet: This tech is the single most expensive tech of the starting ones, with a cost factor of 5. Commercial civs start with this tech. It does not do anything itself, but paves the way to a whole host of other crucial techs. If your civ is going to be going down the "builder" path, then you will want to get this tech early on. For the warmonger, it is not all that important to the early game.
Pottery: Expansionist civs start out with this tech, which also happens to be one of the two cheapest to research in all of Civ3 (cost factor 2). My experience has proven Pottery to be THE single most important tech to acquire in the early game. Why? It lets the player build the all-important granaries! Putting a granary in your capital to build settlers is all but a mandatory step for high-difficulty games. I almost always start my games out by researching this tech at max rate. This should be priority #1 for research in all but the most unusual games.
The Wheel: This tech is very unusual, in that none of the civ traits match up with it. It is, however, a starting tech because the Japanese alone of all civs start out with it. Because of this, if you are playing as the Japanese you will start with a monopoly tech that no one else has, which can be traded for a very high value. The Wheel itself is one of the more expensive starting techs, with a cost of 4. The big thing that this does is reveal the location of horses resources on the map - which the AI civs can see at all times, of course. If you are going to be doing a rush with horsemen, you will want to get this tech and secure a horses resource as soon as possible. It is also one of the techs that the AI civs sometimes ignore, so you may be able to research it yourself first and trade it around for a great deal of value.
Warrior Code: All Militaristic civs except Japan start with this technology, which has an average cost of 3 compared to the other starting techs. It allows the player to build archers and opens the way to horsemen along with The Wheel. If you plan to do some very early fighting, then this is the tech for you. If not, it has a rather low priority in the larger scale of things.
Ceremonial Burial: This is another one of the cheapest techs in the game, with a cost of only 2. All religious civs start the game with this technology. Ceremonial Burial is a cheap and necessary tech that all civs need to have if they want to get some early culture in their cities. Sooner or later, you WILL need this one. Either research it or trade for it early in the game in order to get the ability to build temples.
Iron Working: This tech, the first one of the second-tier technologies listed here, is one of the most important ones in the ancient age. It first shows the location of iron resources on the map, and secondly allows you to build swords when iron is hooked up. Since iron is probably the most important resource in all of Civ3, finding and securing a source of iron is extremely important. You should also keep in mind that swords have both the highest attack and defensive values of the ancient age, to say nothing of the civs that have a sword UU. This is a very high-priority tech even for a peaceful building civ. It is quite expensive though and will take a while to research; the AI also tends to beeline for this one and it is hard to get to it first (at least on the higher difficulties). Trading for this one would probably be a good idea, if you can get the AI civs to part with it.
Mathematics: This is kind of an odd tech, serving as a bridge of sorts between several starting techs and the important ones at the top end of the ancient era tech tree. It gives you access to catapults, which are not that great unless you need to defend against a much larger attacking force. But Mathematics has one great thing going for it in high-level games: because it does not do anything special, the AI civs will frequently ignore it and you can research it first for huge trading opportunities. It's kind of sad that this is the best thing I can say about Mathematics, but it is. Running a 40-turn min science gambit on this is often a good idea.
Writing: Another hugely important ancient age tech. Writing is another expensive ancient age tech, but it opens up a huge range of diplomatic options. Writing allows you to found embassies with other civs, which opens up the option of signing right of passage (ROP) agreements and military alliances. This tech also opens up a number of research paths to other important ancient age techs. If you are playing out a peaceful ancient age - or if you're going to fight a war and want to sign up an ally - get this tech as soon as possible. Writing should have a high priority on the list of what to get.
Mysticism: Not much to say here. It allows you to build the Oracle and that's about it. If you are beelining to Monarchy, you'll have to pass through this tech first. I can't think of too much else to say, other than the fact that the tech after this one - Polytheism - can also be a good choice for a 40-turn min science tactic.
Code of Laws: This tech opens up the all-important courthouses, and helps pave the way to The Republic. Courthouses are not that important in Despotism, but they are one of the most important city improvements to have as the game progresses and you develop your civ. This tech is not that crucial to get right away, but it will be very important down the line.
Literature: A very, very important tech to get if you plan to be doing any research yourself. Obviously, it is even more important for Scientific civs to get in order to build their half-cost libraries. Even better is the fact that since this is an optional tech the AI civs often ignore it, and you can sometimes get it yourself first and sell it for lots of money/techs. Only true warmongers will want to ignore this one. Oh, and you can build the Great Library with this one too. (You can see from that desciption just how important I think the Great Library is.)
Map Making: This tech opens up a number of possibilites. First of all it allows you to build harbors, which can be all but necessary on maps where there is little food and most of it has to come from the sea (see my Epic18b game for an example of this). Map Making also allows you to build galleys and travel across the sea, which may be extremely important on archipelago maps. If you can make contact with other civs first, you will have a huge advantage on the diplomatic front. This tech also allows you to trade world and territory maps with the AI civs; often getting this tech is when the map of the world is finally revealed for the first time. Finally, it allows you to build the Great Lighthouse, which can be a gamebreaking wonder on some maps. Only on Pangea maps is this tech not that useful.
Horseback Riding: This is the tech that lets you build horsemen; it opens up no other techs and does nothing else. Grab it if you plan to use horsies, otherwise you can safely ignore this one.
Polytheism: This tech is one of the ones on the path to Monarchy. It's real value though lies in the fact that the AI civs almost always ignore this one, and it can frequently be researched first through a 40-turn min science move. It gets lumped in with Mathematics and Literature in that regard. Polytheism itself does not actually do anything however.