Either the indexing isn't OK or the chainring is so worn that the new chain is slipping round it. This happens when you fit a new, unworn chain to an old, worn chainring because the teeth are so worn that the chain will only be contacting the first tooth in line rather than spreading the load over all the teeth. A worn chain will have stretched and was probably working OK with the worn chainring.
Typically this kind of chain slip happens when you stamp on the pedals when setting off.
Define slip. Is the chain skipping between gears or is it skating over a sprocket?
OK, that's skipping then, but it helps to identify exactly the right symptoms otherwise you get everyone on a wild goose chase with advice that worked for their grannies and aunts but won't help you in the least. Kinda like the lady complaining about a funny noise under the bonnet.Hard to say for sure, but I think it's skipping between gears.
So probably the indexing.
Weird thing is, I've indexed my gears plenty of times without this issue. Maybe the mech is bent or something.
Might say sod it, and take it to the LBS and let them deal with it.
When was the last time you changed the cable outer where it curves round to the rear derailleur? These need changing at least every six months to maintain good clean shifting and more often in wet weather or on a bike that gets washed a lot. While doing that, be sure to clean the inner cable and pull it through your fingers wetted with a little waxy chain lube. You'll be amazed at the manky rusty condition of the steel reinforcements in the cable outer, which is what causes sluggish and inacurate shifting.