Is it better to start with Wheat and Ore in order to rapidly upgrade settlements into cities, or is it better to start with Wood and Brick to build roads?
As a counterpoint to ICodeForCoffee's answer: I much prefer having access to brick and wood on initial settlements.
Initial brick and wood allow you to get to more positions on the map, to increase your ore and corn income. Initial corn and ore allow you to build cities, which get you... more corn and ore.
If you start on corn and ore, you can find yourself locked out of the other resources (brick and wood) much more easily than if you start on brick and wood.
I would suggest trying to start the game by diversifying your resources and numbers. While I've started games without any access to brick, it usually limits my ability to expand, and keeps me from building out as fast as the other players. How you start affect what your going to do. You'll also have to select sites that are available, since no matter what order you place your settlements, you never get to cherry pick the two best spots on the board. Grain and Ore help you build cities, but you still need to get more settlements you can grow into cities.
I prefer having access to grain, ore, and sheep on my better numbers, and having access to wood and brick on my slightly less common numbers. I try to diversify myself through expansion, although I will settle for multiple instances of a resource if the numbers are good, and there is easy access to a port for those resources. This is a game strategy though, and while you can win this way, it's difficult.
For a beginning player, I would suggest making sure you have access to brick and wood. It helps you out immensely and doesn't make you feel stalled in the beginning part of the game. I like getting access to a 6 chit or an 8 chit with different resources for each of my initial settlements, but only if the settlement locations I'm taking look useful. Definitely avoid building your initial settlements on the coast if the location doesn't have a port. You're reducing the number of resource tiles you have access too, and don't get a port in return.
As a note, I asked a very similar question when we first started the beta for the site, How do you place your initial settlements in Settlers of Catan?
Once you've played your settlements, I suggest first getting new settlements before going to cities and development cards. Definitely though, if you have the resources to build a city, do it. While early expansion is good, using your resources before the robber comes and reduces your hand size is a much better strategy then throwing things you want away.
It is a mistake to choose one strategy. The nature of the game - with the random placement of tiles, the assigned numbers, and the locations of ports - requires a player to adapt to the current situation and apply any number of strategies. Another variable is the order you draw to place your first settlement - if you are first, it might require a completely different strategy than if you are last - depending on the settlement locations that are available to you.
Both strategies can be successful - as discussed in this question your strategy should be based on the initial settlements you are able to get.
Roughly, if you have easy access to Brick and Wood, going for long roads and many settlements can be a good idea. On the other hand, if you have easy access to Wheat and Ore, you'll want to emphasize upgrading your settlements to cities.
Wheat and ore.
Your first city gives you far more dots of production than your third settlement. A starting settlement can have thirteen dots of production, while your third settlement rarely breaks seven dots.
After you get your first city or two up, you can trade your valuable city materials to other players for clay and wood. 1-1 trades are pretty common, especially for wheat, but your production is so strong that you can afford imbalanced ratios (say, 3 ore for 1 clay and 1 wood).
Do be careful you don't get boxed out completely---you usually need to build two more settlements over the course of the game, if only to upgrade to cities for the VPs. If you're conservative about your initial road placement, this isn't hard to engineer, but it is a danger.
Building a city before building your third settlement does bring in the advantage of more income. If your three target junctions are 12, 10, and 7, a settlement and city bring you income on 34 pips, while three settlement bring you 29 pips. BUT! By building a third settlement
a) helps to limit your opponents' options, by occupying a vacant junction, and by building two additional roads, which limit opponents' road building opportunities b) often increases your variety of resources c) contributes to your future effort towards longest road d) offers protection against the robber by making less juicy targets. Not only do you lose, say only 5 pips versus 10, but you may avoid getting the robber entirely when someone else has a city (a bigger target) and you don't. e) probably brings you closer to a trading port, which can be critical in the middlegame.
Overall, I've had much better luck with the third settlement strategy rather than the 'city as soon as possible strategy'.
Wrt another strand in this thread, I've had excellent luck going after development cards right away. Having several cards often makes you immune to the robber--either you can move it if someone puts it on you, or knowing (or believing) you can move it means other players often pick on someone else. Second, and obviously, largest army means two points it's great to have. Third, you often pick up a victory point card or two. Fourth, you often get back the resources you spend, through road building and monopoly cards, or in resources that diverting roober helps you gain.
I don't think resource cards are the best strategy, but they definitely shouldn't be sneered at, a steady stream of them can help you cripple your closest opponent, and if you find yourself with limited mobility it can be a very viable way to win.
Development Cards! These can seem less rewarding in terms of sustainable resource generation, but if you find yourself without road components, buy development cards! Knights can give you some control and protection along with 2 VP's, and the other cards can help you out quite a bit if well-used. Playing Monopoly after a big score of ore or brick can be very satisfying for you, and demoralizing for the other players.
Diversify! Don't compromise on starting with high numbers and 6 separate tiles, unless you are getting two of a 6 or 8, or a port for an item you are set to get a lot of. You're also much better off with a spread of numbers: if you can, sit on a 6 and an 8 instead of two 8's. Spread out with settlements if you can before upgrading to cities. Ore is, for this reason, okay to miss at the beginning.
Generally, you should go for the colonies. However, you have more option if you still get all resources, just trying to have more frequent number for wood and brick.
The good colonies spot may be gone fast. If you first upgrade your city, they might not be there. Also, more colonies can bring complementary resources when cities when only gave you what you already have.
There is a special case, when the wood or brick all have extreme number (2,3, 11 or 12), it will be hard for all player to get road and colonies. In that case, you should first invest in development cards, which will allow you to get an edge in development.