In terms of speeding up the game - you could opt to reduce the cost of trading by 1 card universally (i.e. 3-1 directly with bank, 2-1 with a general port or 1-1 for a dedicated port).
Settlers of Catan is one of those games where there exists an initial learning curve. It is not a long learning curve but it does span 1-2 games to get a basic feel and a further 2-4 games to gain a degree of innate familiarity.
(It takes time to look blankly at a hand of resource cards, often looking towards the build card for your purchase/ build options - and not sure which options are going to give the best results)
As the innate familiarity sets in so too will the length of each player's turn become shorter - significantly improving the flow of the game. Players will hope for the resource cards they need to fit a planned course of action before their turn even comes.
Of course things get more lengthly if you introduce a variant like Cities and Knights. If you see that a variant is taking the fun out of the game by making it take too long then either play the variant less frequently or try any of the tricks already mentioned in the answers received.