In spite of not having any dice, and players being allowed to choose what cards to take on their turn, Ticket to Ride has a significant amount of luck. This means that no strategy is guaranteed to win all of the time, or even most of the time.
But if you want some basic Ticket to Ride strategy, here are some tips on how to win at Ticket to Ride.
DRAWING TRAIN CARDS
The most basic action in Ticket to Ride is drawing train cards. Your other two options are drawing additional tickets and placing trains, but you will be doing that slightly less often. On your turn, the most frequent thing you should be doing is drawing cards.
This is especially true towards the beginning of the game, when you're trying to figure out what options you have. With only a few cards in your hand, it's not easy to determine which color tracks you'll want to lay down, even if you know what tickets you are trying to fill. Until you have more cards, you don't know which color you'll need your wild card locomotives for.
That's why drawing cards early on is an important strategy. And unless you have a good reason not to (such as later in the game when you need one specific color), you should always draw facedown cards from the deck. The reason for this is simple: You might get a wild card. Drawing a face-up wild card takes your whole turn, so if you can draw two cards from the deck on your turn and one of them is wild, you've gotten a free extra card you wouldn't have gotten from taking face-up cards.
Once your hand is more filled up, it will be easier to see what colors you have a lot of, what colors to use your wild cards on, and what colors you need so badly that it's worth picking them up from the face-up cards. Until then, keep drawing from the deck!
DRAWING ADDITIONAL TICKETS
Contrary to what you might think, you don't always need a lot of tickets to win the game. Yes, completing more tickets gives you more points, but failing to complete your tickets will lose you points, making it more difficult to win.
One common strategy is to take a turn to draw tickets near the beginning of the game, to see if you can match any new routes to the ones in your hand, ideally finding some overlap. Usually a better strategy is to take a few turns to draw train cards first, and then if you want to take a turn to draw tickets, you'll have more information about what color tracks you might be able to play on before choosing what tickets to keep.
The goal of the game is completing tickets, which means once you have completed all the tickets in your hand, you should be doing one of two things: Either trying to end the game as fast as possible, or drawing more tickets to complete. If the game continues but you complete no additional tickets, you lose ground against the other players.
Placing trains on the board is a balance between not wanting to move too soon, and not wanting to move too late. If you play on the board too early, other players can easily see where you are trying to make connections, and place trains to block you.
On the other hand, if you wait too long to play your trains and just draw cards for a dozen turns, some of the key small connecting routes (like Atlanta to Nashville) will disappear before you get a chance to take them.
As a good rule of thumb, take a few turns to draw cards at the beginning of the game before playing any trains. What you want to avoid is playing one early connection on the board, and then having to draw more cards for three turns while other players block that connection. You should instead draw sufficient cards that you can take a few turns in a row to play trains, so other players don't have as much time to block you.
If a short grey connection is part of your route, build that first. Other players won't necessarily be able to save up enough cards for a 6-length colored connection you need, but every player can easily play on a short grey connection, so those are the most urgent ones to grab.
Finally, try to build outwards from your original connection to form an unbroken line of connections stretching from east to west (or west to east). This cross-country line will not only give you more points (especially if you can get longest train), but will also make picking up and filling additional tickets much easier, because any two cities you need to connect probably aren't far away from somewhere on your line.