Bicycle Tutor

I've replaced the chain on my 7 speed bike and was told in the bike shop I had to replace the rear cassette? Is this true?
The new chain seems to be causing some slippage when I'm riding (on flat and hills) and starting to pedal the chain is skipping and not getting the bike to move.
I've got a new middle chainring but even using that to start off with also produces the chain skipping.

There is a fair chance that the cassette is worn. Try each sprocket in turn. Some may be O.K. and others will slip. They will usually only slip under pressure when either setting off or climbing.
If any slip then you will have to replace the cassette. It's wise to check your chain every so often to see if it is worn.
See Alex's tutorial on chain checking:-
http://bicycletutor.com/chain-wear/