Did as advised (pulled the cable taught) but there still doesn't seem to be quite enough tension to pull the derailleur up far enough.
I was doing it by hand - be ok to use a pliers to pull and get that extra bit of tension?
You'll struggle to get the derailleur to move properly by pulling on the cable, you'd be surprised how much tension they offer. While you can see the slack in the cable, give your derailleur a push from behind (from the left pedal side of the bike). If there's movement in the derailleur I'd say you could relax about it being anything serious. If you know where the cable anchor point is, set the front gears to the smallest cog. Loosen the anchor point, pull the cable from the end until there's no more slack, tighten the anchor tightish. You should be able to get the derailleur to move by using the shifters to change as normal. If the derailleur moves smoothly, you've ruled out cable problems within the shifter housing. If the derailleur won't budge or it feels notchy, you could have problems in the shifter housing and you'd be best leaving it til you get your service.
If the derailleur moves smoothly once you've taken out the slack, all you should need then is the gears setting properly.
Yes the shop could sort all of this out in one go, but who's to say what they may try and 'blame' you for. Best to try and get a handle of what's wrong, then you can go in more confident that you won't be ripped off. (Not all bike shops are bad... might just be where I live... except my shop of course...).