Sounds like you haven’t pulled the shifter cable quite far enough through the deraileur clamp when you put it back together.
Drop down to the smallest gear, loosen the cable clamp and pull the cable through a few millimeters and retighten. Then run the chain up through the gears. If you get onto the biggest cog without difficulty and can still get to the highest gear, then you have adjusted the cable within “the zone”.
Now use the small adjuster screw that the cable passes through into the deraileur. Spin the wheel and change up and down the gears until there is no skipping and the changes are slick and smooth. Look closely at the position of the jockey wheel tracking in relation to the selected cog as you turn the screw. The jockey wheels should align with the selected cog.
There are two screws in the deraileur to adjust the limit of its travel both inwards and outwards. All these do is prevent the chain coming off the cassette. Identify which screw does the inner and run the chain up to the large cog. Adjust the screw so that the deraileur can’t shift the chain into the spokes, but will stil change up and down between the adjacent cog. Repeat the process with the other end stop screw for the outer (smallest) gear. Again. observe closely as you rotate the screw. Turn either way until you see movement. Then back it down to stop the chain jumping off the end of the cassette.
This should sort it out. A ten munute job at most.