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The special destinations cause problems as those passengers take up the space of the triange/circle/square passengers to trying to deliver those efficiently is important. You may want to make a special station be the common station between lines, especially once more squares appear.
Try to use loops, but you can't always do that. Notice it will be sometimes dump a, say, circle passenger, if another line can get it to a circle more efficiently, but then that burns space in the other line which is a pain. Ideally you want the shortest distance for passengers. It is not always obvious.
i tend to go for carriages and keep lines and tunnels to a minimum. With a carriage it means you get an increase in capacity of two trains worth, instead of one. But you may have to take lines and tunnels to make your network work.
Then, the population explosion happens and it gets very difficult. At that point i will pause, release, pause, etc, and move around trains to empty out busy cities, hoping to push my high score a little higher than last time.
The other strategy is to keep a train back for emergencies, when a city is being overloaded. Pause frequently as danger can come out of anywhere.
These are my thoughts.
I mostly think in terms of visit frequency (for a station) and length-on-trains (a passenger remains on a train, occupying space, until they reach their destination).
1. Too many stops of the same type in one line is a killer. You don't make it to the 1000's in Hongkong if you line up more than two of the same type on a regular basis.
2. If you make a loop you basically make the visit frequency equal for all stops on the line, no more, no less. Effectivity changes if you let trains run in opposite directions. This is not necessarily better!
3. Stops in the middle are visited more evenly spaced; stops at the end of a line are visited twice in short sequence. A square or triangle at the end of a line is therefore much much better than a circle.
3b. Better yet: a square in one direction, a triangle in the other.
3c. Better still: squares and triangles in both directions!
4. Special stations are mostly game-killers for keeping your trains occupied for longer amounts of time, or for needing detours through several lines.
Basically, two solutions for an overloaded system is to reduce time-on-train and increase visit frequency. That should cover most of it.
For river crossings, my loops through a river have been more effective than single tunnels, unless the singe tunnel lines serve 2 or 3 stations.
I try to build my loops in a sequence both of you describe. Ex: Circle, Triangle, Square, Circle, Triangle, Circle, etc. Alternating stations seems to be more effective. If there are two of the same station in a row in late-game, the first station will completely fill the train, leaving the second station untouched, which leads to a rapid game-over as the second station will never be serviced,
I usually try to split up the special station between lines if I can or even have a couple lines per special station.
When you build your lines, you know how the angles can sometimes change a lot depending on how you move your mouse? Well, I have noticed that trains will always need to slow down and stop at stations which are serviced by an "acute angled" track. That is: when a line goes through a station with an angle of less than 90 degrees, the train will always perform a slow down and stop at that station, regardless of if there are any passengers waiting there or not. So if you have multiple acute angles along a track, it can become really sluggish and backlog very quickly as the game speeds up.
Since discovering this, I always try to make sure that when my track has to do a turn, that the turn is at an angle of 90 degrees or more. This allows any trains on the track to zip around empty stations quickly, rather than stopping in places it doesn't need to. If I absolutely have to create an acute angle to reach on oddly placed track, then I won't dwell on it too much - but also keep an eye out for other spawning stations which will allow me to get the line back to 90 degrees or more at all turns.
One other little trick is one I found on a guide online. Circular lines are very efficient when used correctly, and even more so when you have multiple trains and carriages on the same track. However, when you add those extra trains, which way do you have them going? If the trains are all going the same way, each of the additional trains is programmed to leave a "buffer" between the one in front of it. I think the distance is one or two stations. This has a negligible effect on a forward-and-back track, but on a circular line it can often mean that the trains at the back are constantly stopping and waiting at stations un-necessarily while the train at the front moves along.
However! This is completely voided when you have two trains going in opposite directions, and they'll zip past each other without a care in the world. So whenever I build a circular line, once multiple trains get involved I always make sure they're travelling counter-clockwise to each other. You can force this to happen by breaking the circle at one point until both trains are heading in opposite directions, and then putting the two ends back together again.
I too am quite a casual at this though, so my "strategy" (lol) basically consists of circular lines all over the shop, and an early-game "island line" which services all the stations at one side of any large body of water until the map is large enough for that area to warrant more than one line. The island line will then cross the river and link up to one or two stations on the mainland side, which I then connect as many other lines to as possible and turn into interchanges.
Hope some of this was able to help!