It will vary and some will probably have better ideas but this has been good for me up to the medium level of difficulty; I haven't played at higher levels on Civ3 yet.
1) Use your first worker as an explorer and at least locate barbarian camps.
2) Found your city immediately. If there is a huge difference by moving 1-2 turns fine but you are in a stage where a tiny delay can have a huge negative impact.
3) Build warrior. Use to explore NEAR the city. It is unlikely that you will be attacked during the first few moves of the game.
4) Build a second warrior. If you try a settler it is usually going to result in the delay that you mentioned which is a total waste.
If you are lucky, by now you have located some other resources through goodie huts AND the enemies city. Keep your warrior between the enemy city and your city and you can intercept their scouts coming at you, covering your home base.
5) Build a third warrior. Make this one stay home.
6) Build a settler. Should have no problems now.
7) Build a barracks. Start churning out vet warriors. Ditto from your second city. (or archers or horsemen if you have these available)
Note that you haven't built a spearman yet. You should have at least 3-4 warriors, half of them veterans, and your nearest rival city scoped out. If you have a worker left, use it to start building a road toward the enemy city.
8) Crank out vet warriors and attack and take the rival city. Use good tactics and effectively use geography -- for example make sure you are not attacking across rivers!
You should now be able to grow most of your civ during these early times by repeating the barracks/warrior (or other offensive unit) formula.
There will come a point at which you have created sufficient space to grow in a more constructive fashion, perhaps after you get 5-6 cities or depending on circumstances even 10-12. Your veteran offensive forces will make an excellent defense that will carry you into the early middle ages at least. If you are really lucky you can do in two rival civilizations, and take one or two early wonders which someone else has created for you.
Then you can make spearmen to hold what is really worth holding. They are a good investment after the early conquest stage because they will upgrade in the future. Warriors do not.
