I have a Roper top loader washing maschine. It runs the cycle but will not move on to the next cycle on its own. I have to stop it and move it manually to the rinse or spin cycle, pull the knob out again and it will run the selected cycle. What could be the problem and how could I fix it?

Hi from retired Englishman in SW France,
(no promises but try this!)
I would gently suggest as a first step that you clean the drain motor/pump lint tray/filter. So many 'faults' are caused by blocked filters and many machines will not even go to rinse or spin or release the door lock if there is any water still left inside.
Here are my complete and generic guidance notes for checking both the drain motor filter and the drain motor ;-0)
If you are unsure where the filter is, it is often behind the front lower (kick) panel (or in some machines at the rear in an awkward position!). These panels are either held on by (plasic) screws or will just pull off. Most filters are held in by a round cover/knob. Sometimes the filter will not come out with the cover and you will have to persuade it to come out to play.

REMOVE POWER AND ENSURE YOU HAVE A WATER RECEPTACLE/RAGS TO SOAK UP THE WATER WHICH IS BOUND TO POUR OUT (alternatively use a wet vac as sensibly suggested by VinnyB1234!)
It's surprising what people have found in theirs! It is always worthwhile checking it on a regular basis.
Now- whilst the filter is removed it is normally possible to see the drain motor and ensure it is not jammed by a foreign object such as a needle or a toothpick. It should be able to be turned by hand/finger- the power IS OFF!
At this point it is also possible to test if the motor works. No clothes or water in the machine, select a spin cycle and put the power back on. You should be able to see the motor turning at some point, possibly straight away. Using something suitable like a glove see if it is easy to briefly slow the motor down. If it is easy then the motor is malfunctioning.

If this has assisted you please consider a 4 thumbs up for the rating.
Thanks and good luck!
John C