The placement of your initial settlements is very important.
Your initial settlements (along with the first couple rounds of dice rolls) have a large impact on your game, affecting whether you get off to a quick start or lag behind. Some things to keep in mind when placing your initial settlements include:
- Probability of numbers: All else equal, a space bordering more dots is better than a space bordering fewer dots.
- Diversity of numbers: In general, starting with, e.g., 4-5-6-8-10-11 is going to give you an easier game than 4-4-6-6-10-11, since you will receive cards on more dice rolls, which then helps you build and/or trade more consistently, and helps you avoid having too many cards in your hand when a 7 is rolled.
- Mix of resources: All else equal, it's better to have direct access to more resources.
- Subpoint: In general, wheat is important at all stages of the game, sheep tend to be slightly worse than other resources (due to lower need and relative abundance). Note that the usefulness of resources changes as the game progresses.
- Scarcity of resources: However, there will usually be one resource that is scarcer than other resources. Having access to this resource can give you an advantage in trading. A simple way of evaluating scarcity is to count the dots on each resource.
- Don't ignore ports: Sometimes, ports (especially a 2:1 port) can increase the appeal of a spot. A wood-wood-__ spot next to a 2:1 wood port may be more appealing than one with three different resources. (Of course this depends on many other factors.)
- Don't count on being able to expand to the good spots -- or the interior of the board: In general, it tends to be harder to expand into the center than towards the edges.
Make sure you consider the consequences of your trades
Sometimes a trade that in other situations would be relatively even will give one player a large advantage. Trading a card to a player so that he can build a settlement on a wood port before his neighbor can might be a fine move, even if it upsets his neighbor a bit. But if that player has settlements on 8-wood, 6-wood, and 4-wood, it might give him too much of an advantage.
Consider the robber placement
Placing the robber on a player with an unplayed development card might not be the best move, since it is relatively likely that the player has a soldier to move the robber away.
Don't neglect development cards
Soldiers (the most common card) provide you with a resource, deprive an opponent of a resource, free your number (if blocked by the robber), block an opponent's production, and can potentially provide 2 VP. Road Building, Year of Plenty, and Monopoly are nearly always useful, except perhaps for a late-game Road Building. Victory points are of course useful. But make sure to evaluate the potential drawback of using (in particular) ore and wheat for a development card instead of a city.
Longest road is nice -- when no one else wants it
In my experience, many beginners go for the 2 VP from the longest road. Roads are easy to build since they only require two resources, but also provide the fewest benefits. The best way to claim the longest road bonus, in my opinion, is (1) in the mid- or end-game, after establishing a good resource engine with multiple settlements (and therefore multiple resources / numbers), (2) when other opponents are not focused on attaining the longest road, and (3) when you have an excess of wood and brick and other players aren't willing to trade at a good rate for either. Otherwise, it's probably better to focus on building settlements, cities, and even buying development cards. In short, don't just build more and more roads because they're easy to build.
Other resources
https://boardgamegeek.com/thread/137013/tips-beginners-catan-dummies
https://boardgamegeek.com/article/7450280#7450280
