What you didn't make clear, and I suspect from your later posts, is that the bike was secondhand? If so, there is only so much the bike shop will do before you need to replace components. If you have replaced the chain, then I suspect it's jumping because either the chainrings or the cassette, or both, are also worn. As a chain stretches it wears the cassette in such a way that a new chain will not mesh properly with the shape of the teeth.
£50 quid for brakes including a £25 servicing charge isn't bad. They'll charge you retail price for the blocks and cables, and an hour's labour for the fitting. Brake pads and cables would cost you about a tenner if you shop around and fit them yourself, but if you don't learn to do it then you will be at the mercy of people who have a living to make.
The frequent punctures may be down to knackered rim tape. Take the tyre off and look to see whether there are any areas where the spokes are exposed.