Having made my own transition only 8 months ago, I am happy to share my experience with you.
First of all, congratulations on a wise choice. You won't ever play Civ2 again, because Civ3 is better in almost every way. However, the transition was extremely frustrating for me and you may very well feel likewise if you try to win the same way you did in Civ2.
General principle: nothing that made Civ2 easy even on Deity still works in Civ3. There is no way to catch up in tech with spies, to build wonders collectively (caravans being gone), to block CPs' expansion with a few well-located units or cities (no more ZOCs) and to grow cities in no time with WLTKD ("We love...").
Also keep in mind that combat has changed a lot: stacked units defend themselves one by one, and when you destroy the last one, the victor will occupy the location of the defeated, making him vulnerable to counter-attacks. Hence you can no longer defend a city forever with 2 good defenders and 2 attackers. Furthermore, the AI won't make futile attacks forever but may turn around your strong position. They are indeed (somewhat) smarter.
The economics have changed too. The new production to gold change rate (4 gold for 1 shield instead of 2) makes "buying" improvments much more expensive.
Politics have changed too. Since you can now declare war under representative govs, and troop maintenance is now paid in gold instead of shields, choosing between Monarchy and Republic is now essentially a mere budget issue.
Monarchy is no longer the great gov it used to be. Forget rushing towards it as you probably used to: it is now much further away and not worth it, unless CPs give you part of the tree tech (Mysticism, or Polytheism) and you plan to keep at war all the time.
Democracy isn't worth it either, unless you have a huge empire. On a standard map, you'll gain only about 2.5 to 5% gold compared to republic, and this bonus certainly isn't worth researching two non-necessary techs (Press and Democracy), not to mention the anarchy period.
If you have neighbours, cultural influence is essential. Temples now only make one content face instead of two, but early temples are immensely powerful in the long run.
Finally, learn to use pre-builds for Wonders, and focus on essential ones (Leonardo's no longer being one of them!).
That's the most essential things I had in mind to advise a beginner.
Anyway, try to play Civ3 with a fresh perspective, because it's almost a new game. It is much more different from Civ2 than SMAC was, for instance.
However, I don't advise reading too much about the game. Try to discover your own way of playing it. You'll enjoy it, for sure.