General Strategy

1. Try to maximize the chance that any given card being drawn is both playable and contributes to your score
2. You will frequently find yourself in the situation where you have to "guess" which of the remaining cards is next to determine which station you should draw toward. But it's never truly a 50/50 decision - there are tie breakers to be taken into account. Which outcome has higher upside? Does one result include more red card(s) which are guaranteed to be drawn eventually vs blue cards which may or may not be?
3. Aim to draw each route such that you have many stations in one district and then at most one station in any other district. Drawing 2 stations in 2 different districts means you have effectively thrown away a station losing you an opportunity for about 5 or 6 points. A good route where all blue cards were drawn should score at least 5x6
4. In the event that you are forced into taking a bad station (this will happen) try to take the station that has the highest opportunity for future upside - usually an interchange with a later route. In some cases not drawing the station at all is the best option because it can get in the way
5. Every route start point has 8 adjacent stations, 2 of each type. Keep this in mind while drawing earlier routes. You should avoid making both stations of any given type inaccessible from a start point
6. Consider the trade-offs of drawing a "long" connection between stations. This will limit your future options, but may let you score more points now. It's usually obvious which color route is most effected by drawing one of the long connections. If you haven't drawn that route yet, you often should not use that connection

Objectives

At the start of the game analyze the objectives. These can have a significant impact on how you should prioritize drawing each route as missing an objective will usually mean you lose. Break down of each objective:

1. 6 River crossings - there isn't much to say on this one. Usually you should be crossing the river 6 times regardless of how you're drawing, but do keep it in mind and try to prioritize getting to the river a little more than normal
2. 9 Central District Stations - again fairly simple. This one should not be very hard to achieve assuming you're pushing 2 of your routes through the center like you should be
3. 8 Interchanges - this is a bit more challenging. Draw 2 of your routes with the central district as their majority which should get you 4-5 interchanges. Sending a route through one of your future route start points can be good but be careful that you're not ruining that route's regular scoring
4. 13 Districts - this is the most difficult objective. Certain card orderings can make the objective impossible or at least not worth pursuing. Drawing into an outer corner district is inherently sketchy. We want to keep both segments of our route playable as long as possible. But if we draw into the corner, there are now no longer any good stations for that segment (results in a 2nd station in the inner corner district which is almost always 0 points). If you're playing super hard for this objective, you may almost always achieve it, but if you sacrifice more than 10 points from your regular scoring (~2 "missed" stations) then it wasn't worth it.
5. 5 Tourist Sites - plays interestingly with normal scoring. Each tourist site except the central one has a natural route for scoring it. But that forces you into drawing at least 2 routes in its starting district which means you would forgo any central district majority which is bad because the central district majority routes have the highest scoring potential. So the best way to play this is to try to hit tourist sites in non-start districts on 2 of our routes which also draw their majority through the center

Pencil Powers

Each pencil power has a unique impact on your route. You should always aim to use it, but you also need to be careful not to waste it.

1. Joker - this lets you play a little more "loosey goosey". You can take bigger risks knowing there's a wild card to get you out of it. Until it's gone
2. Draw a second station of same type - this is probably the hardest power to use effectively. I usually end up using it more liberally than any of the other powers because you can often find yourself in a position where doubling up any of the remaining stations doesn't actually score more points
3. Railroad Switch - plays somewhat similarly to Joker. Makes it much easier to play into a dead end knowing you can start up another segment on demand
4. 2x Station - the easiest card to play. Note how the math changes making drawing across additional districts more valuable