Ticket to ride strategy 1 - Route stealing
If you click the ticket to ride map on the right, you will see that the green player first took the route between "Seattle" and "Helena". the second ticket to ride route he took was "Duluth - Chicago". It's fairly easy to assume that this ticket to ride player will later try to connect those two route by taking the route connecting the city "Helena" and "Duluth". So what you should do at this point is try to take possession of this route for yourself, but not at all costs. Take a good look at your TTR destination cards and try to find one that need that route. If none of your destination card needs it, then you should forget about taking it unless only a few train card are needed. That last ticket to ride tip lead me to my next strategy.
If you look at the ticket to ride map, you will notice some route junctions that requires very few TTR train card to buy. The strategy here is to buy some route for "nothing". What I mean by that is that you kind of waste some resources just to prevent another Ticket To Ride player from getting the shortest path to his destination city, forcing him to make detours that will cost him in the long run. In the US map of ticket to ride, you can find many strategic routes that you can take to acheive this strategy. For example, if you take possession of the route between the cities of "Houston" and "New Orleand", you will force you opponent to go around this route and lose a lot of train card and turns. This is only one part of the cheap route buying strategy, the other part consist of taking the route that will pay the most.
My previous example shows how to capture cheap route but might not help that much since most of the route can probably go through the center of the map. My suggestion to improve this ticket to ride strategy would be to try taking the cheap center route. Routes between cities of "Kansas City" and "Saint Louis" (both blue or pink one), the route between "Toronto" and "Pittsburgh", the TTR route between "Nashville" and "Atlanta" and finally (even if it's more expensive) the routes between "Salt Lake City" and "Denver" (both red or yellow). I took the last route because it's one of the few chocking points on the west coast. Key routes are identified by a green check mark.
Ticket to ride strategy 3 - Selecting destination cards
Has mentioned in the ticket to ride rule book, at the beginning of the game you have the choice to keep all destination card or to discard one. When I play ticket to ride strategically, I won't keep three destination card if they don't share common routes at all unless they are all very short distance cards. So try to keep destination card offering the best alternatives when choosing paths. That means to try not keeping destination that only have one or two long route to get there. "Calgary - Montreal" is a good example here. Always keep in mind "ticket to ride strategy 2" because someone else might just use the same strategy.
Good luck and feel free to share your best Ticket To Ride Strategy!
Ticket To Ride introduction
An introduction to what you will find if you play this "soon to be a classic" board game. A general description of the game play and elements.
Ticket To Ride Maps and Rules
The reason why this board game is great is in good part related to the fact that the rules are different for each version of the game. You play the american version for a while then you buy the European version to play an all new set of rules. And for the map lovers, check them out here!