Which order do I play the Civilization IV expansion packs and the main game?

Solution 1:

Civilization IV: Beyond the Sword is the latest expansion, this is what you want to play if you're starting out. This contains the main game plus all the changes and additions from the two expansions.

You can play the vanilla game or just with the Warlords expansion, but then you'll miss out on the improvements made in Beyond the Sword. This option is more for players who prefer the old rules, as a new player I would ignore this option.

The Warlords expansion also contains some scenarios. You can't access those if you start Beyond the Sword. If you want to play the Warlords scenarios you have to launch Warlords instead.

Colonization is a completely different game, just using the Civ 4 engine, as you already said, this one you have to start separately.

Just start Civilization IV: Beyond the Sword to begin.