OK - so I'm back.

Since I last posted I have had the bike out quite a lot, and in the main it rides well.

After tightening the gear cable the gears worked well most of the time, but still missed the second cog on occasions - until I found a reference to adjusting the derailleur barrel adjuster. Once I adjusted the tension in the way it was shown on a video the missing went away completely.

However, I now have a second problem which is that the chain sometimes slips off completely when I change gears (almost always when changing from a larger to a smaller cog - and not always on 6-7, sometimes when going from 4-5 or 5-6).

I spoke to a local bike shop who said that they would have to see the bike to diagnose (fair enough) but at a cost of £30 + VAT, to cover 1/2 hour labour, within which time any parts replaced would be extra. Over 1/2hour add £30.
Thats more than the bike is worth, and their parts prices were also way over the top.
Unfortunately they are the only bike workshop within an hours drive of me, so it's back to self diagnosis and repair.

I could replace the chain (clarks), derailleur (Shimano TX55 6 / 7-speed) and cable (clarks) for £6, £9 and £2 respectively - add a further £9 for a new cassette (shimano tourney 7 speed screw on) - a total of £17 or £26.

But my question is -
is there one part that would be most likely to cause the problem in the gear change which sometimes sounds clunky, sometimes is silent and sometimes causes the chain to slip off completely when changing down.