I am new to Travellerspoint and I would like to edit the color of the curves from stop to stop on my map to reflect the type of transportation used (plane, car, boat, etc.).
I would also like to edit the shape of the curves themselves to follow for example a river's bed, etc.
Is it possible to do so?
Yes and no. You can get the effects you are after, but to do so is a bit of a hack, so a bit time consumng. To get the first, you'd actually have to create a new trip for every leg of the trip. Then you can change the colours of each trip as you normally would.
You can edit the curve by dragging the line, but you can't change the shape (it stays a curve basically). So to get a line to follow a river you'd actually have to add stops along the way to get the line to follow those stops.