Adivce on what to do next with crunching / slipping gears
That sounds like the rear cogs are worn, the likely fix is a new chain and set of cogs for the rear (you have to replace them both at the same time otherwise the problem will persist, perversely even worse).

I suspect the bike shop fixed/replaced the stuff that was obviously visibly knackered and hoped for the best.  In truth I suspect they needed to do what they did plus replace the cogs and chain, but they were hoping that wasn't the case because the bill would have been even higher.

In my experience bike shops don't tend to actually ride the bikes so they can miss stuff I wish they wouldn't.  That's one of the reasons I learnt to do my own maintenance.  

In general the trick with bike shops is to check your bike before you sign off on the work, i.e. I ask to ride my bike around to test it before I leave the bike shop (and pay).  If I do find I still have a problem after leaving the bike shop  and paying I go straight back and explain the problem and say it's not quite sorted yet.  They are normally very reasonable and will sort the problem without additional charge, assuming it doesn't require new parts.  

Your best bet is to go back to the bike shop and explain the problem isn't quite sorted and if possible demonstrate it to them.  It will cost more money though, it sucks but unfortunately bikes aren't generally that cheap when it comes to maintenance (compared to the initial cost of the bike).