Beginnings.... When you start a new Settlers III game, always build about 3 or 4 woodcutters, 1 or 2 foresters, 2 stonecutters, and 1 sawmill. You may sacrifice your early stockpile, but these buildings will fuel later expansion.
Expanding. Early in the game, build many small towers, but after you have about 50 pioneers, you should be thinking about crushing them and expanding with pioneers.
Manna. Once your construction industry gets going, and you have enough boards and stone, build 7 or 8 manna-producing centers, and 4 small temples/sphynxes. With this, you will be able to promote all of your soldiers quickly and with ease. Also, your priests can take advantage of the magic.
Maps. In BlueByte's Christmas map, Star, you can locate a lot of hidden gold and other precious materials. The top left end of the star contains a lot of gems, and that can help you win the game. The top center end has a lot of gold bars, and the island to the top-right of the star has a LOT of weapons, gold, wine, pigs, and other cool things.