Why do my car's gears keeping sticking?

Quoted from the Complete Car Care Manual by Popular Mechanics:

“Does the harsh shifting occur only in one gear change…? The accumulator is something to suspect on fully hydraulic and partly electronic automatics. Most hydraulic clutch/band circuits have an accumulator, which is a spring-loaded piston assembly in a chamber, to prevent shift shock. If the chamber is filled with sludge or the piston spring is broken, it could be responsible. If the accumulator is external, you can remove the cover and look inside.
Of course, a sludge-filled or varnished accumulator chamber could indicate a varnished valve body under the oil pan. So one harsh shift now could become harsh shifts soon. However, just cleaning out the external-access accumulator now could be smooth-shift news for a long time.
If the accumulators are on the valve body, the job is more difficult because you’ll have to remove the oil pan. But at least you get a look at the valve body too. Refer to a service manual to see what’s involved in an inspection” (Popular Mechanics 143).

Complete Car Care Manual. New York: Hearst Books, 2008. Print.

You can either ask a mechanic at a shop or look in a book to find out if (a) your car in particular has an accumulator, (b) where both that and the valve body are located, and© how to get at and clean both of these, if they do need cleaning.

Try to check your clutch. It may worn down.

Did the problem start suddenly? does the upshift always happen at teh same speed? It could be the throttle position sensor. The computer can think you are at wide open throttle so it holds the lower gear longer as you accelerate.
It is impossible to give you a full definitive answer.

Do you have enough transmission fluid in your reservoir?

I’m with @brian1946. Drain the fluid and refill, see what happens, it’s the easiest test.

Start a transmission savings account

I second the transmission fluid. What happened to my car (gears kept sticking). Also, hate to suggest it, but your whole gear system may need to be changed out. happened to me too, sadly…

I would take it to a professional, unless you know a lot about transmissions.

I know that certain years of Dodges had bad transmissions (actually, especially bad, since American cars in general have questionable trannies, but I digress) so I am leaning towards a suggestion you may not like.

Repairing an automatic is generally more hassle and expense than it’s worth, so it’s often cheaper and easier to just replace with a rebuilt trans rather than fix the one that is in there now.

You can try a transmission flush first, but I have my reservations. Then again, without actually seeing the car with my own eyes, it’s hard to get an accurate diagnosis.

A fluid flush and filter change would be the first step, and some Lucas Transmission Fix additive would definitely help. It’s probable that the Clutch Bands are out of adjustment, but that would be beyond your ability to fix outside of a transmission shop.
You can do a fluid flush pretty easily on your own but pick up a Chiltons or Haynes manual for your car first, they are not the greatest in the world but they certainly help. If changing the fluids and using an additive doesn’t help then take it to a transmission shop, not a regular mechanic. Replacing a transmission is never cheap or easy, and nine times out of ten it’s not even required.
For Dodge/Chryler transmissions always remember they use the ATF + style fluid. In the case of your car it will be ATF +3 or +4. Do not use Mercon/Dexron of any kind, and try to stay away from “Multi-Vehicle formula” junk