Sounds like you cut the chain a little short.

8-speed chain is good for 6,7,or 8 speed gears.

Skipping could be a lot of problems. Worn chainring, worn rear cog. If it's wanting to change gears in back it could be poorly adjusted rear derailleur.

Look for derailleur adjustment vids online. I like to release the cable, make sure center of top pulley aligns with outside face of small cog (using H limit screw)
Then pull cable and anchor it. Some low-end derailleurs seem to like the cable a little loose, so don't pull super tight before anchoring.

Then shift through gears, adjust cable tension til they work, shift to big cog, adjust L limit screw if needed.

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