so yea, build in the new world or not?
It really depends on your circumstances. If you've got enemy civs breathing down your neck back home, then the effort of colonising the new world probably won't be worth it (at least until you're secure on the home front).
If, on the other hand, you're short of expansion options (eg. if you're stuck in a corner with only a single powerful, but friendly, neighbour), or if you need the health/happy/strategic resources that are available in the new world (or could use the new world ones to trade for the ones you need), then colonisation is an attractive option.
In either case, explore the new world asap, pick up those juicy undefended huts, and try to spot out the best city sites. Explorers with the Guerilla II and/or Woodsman II promotions, backed up with naval vessels (to search the coasts and move the explorers quickly to the huts), are the best units for this.
You have, I think, three main options for new world expansion:
(1) Just take the very best city sites, try to stick to the coast (or build a 'port' city to make the trade connection to that resource-laden inland one), and get your fogbusters and defensive units up and running asap.
(2) Choose an area in which to concentrate (consider resources, city placement, strategic value) and build several cities there. This should give you a smaller border to defend against barbs, makes infrastructure much quicker and easier to build, and makes the maintenence-busting power of the Forbidden Palace more effective.
(3) Try to conquer the whole damned thing. A tough proposition and a heavy investment. Probably only a worthwhile option if you're going for a domination victory and you already have a massive population back home, and so only need more land to win it.
Edit: I forgot to add... If you're pulling in the cash at home, then you can run Universal Suffrage to quickly complete the first vital building or two in your new world cities. Slavery is obviously another option once you've grown the cities a bit. You might even find it best to use a combination of both for ultra-fast building. A Great Artist culture-bomb is another useful tactic, as it can cut down on barb incursions, link up the borders of two or more cities, snag a couple of extra resources, and it increases your land area for score/domination purposes.