Chesterfield has a good train service with 5 trains per hour in each direction for most of the working day. These include direct trains to Liverpool, Norwich, Leeds, London, Plymouth an Edinburgh as well as Sheffield. See a map of serviceshere.
Combining cycling and rail travel is good for long distances. Full size bikes must be pre booked on most services. This can be done in advance on the internet using this website (tickets for you and your bike). Note that few services will book more than 2 bikes. Folding bikes (wheels of 20 inch or less) travel without restriction if folded and covered.
Chesterfield station has parking available on platform 1 for about 30 bicycles. This area is covered by CCTV.
Using a railcard will reduce the cost of train fares considerably as will booking in advance. Fares to London are regularly available from about £8, sometimes even £1 if booked via the Megatrain website. Of course peak time travel will be considerably more.
If travelling to other towns and cities by train without a bike think about buying a PlusBus ticket. They give you unlimited travel on buses at your destination (they can also be bought for travel in Chesterfield) with some restrictions. Contrary to the advice often given at Chesterfield railway station Sheffield PlusBus IS valid on Sheffield Supertram.