2002 Buick Century shifting problems

I have a 2002 Buick Century with a 3100 engine. It has a shifting problem and I have maintained this car very good all fluids and filters have been changed as needed and on the correct intervals. When you drive it regular between 20 and 50mph there is alot of hesitation and its seems as it is having trouble shifting but it does shift and it does not shift hard. When you have to pull out in the road and have to accelerate rapidly it does not shift into second until about 40 or more and into 3rd at about 60 or more; it makes the engine go to a very high RPM. This car does not have a RPM indicator as you well know. Also; the car gear indicator light has gone out I have to feel where I am going into gear, I dive in the overdrive selection as suggested in the manual and I know that is right past nuetral; so I am in the right gear. This light being out also is the same light that shows the odometer so that is a small problem as well. I was wondering if you think the shifting problem is in the transmission or a sensor of some sort i.E: speed sensor etc.