multi_engined, from your recent posts I take it you have a PPL or are just about to attain your PPL. Having said that, I am not gonna answer your first question about piston engines as you should already know why it goes from 0-1000 RPM almost instantaneously.
You also shouldn't worry too much about turbine engines now, when you study ATPL Systems, you will go over turbine engines so many times it will make your head hurt.
It is suffice to say though, without going into too much detail, that turbo-fan/prop spins because it is directly connected to the turbine at the back of the engine. The turbine at the back of the engine spins because hot air from the combustion chambers flow across the turbine disc thus causing it to rotate.
When the turbine engine is started, it takes time for the turbine (at the back) to accelarate from zero to idle because fuel isn't added to the combustion chambers instantly like it is in a piuston engine. Normal start-up should be no more than 25-30 secs, and if its any longer you probably are dealin with a hung start. Again there's a lot involved in turbine engines, and you'll appreciate the sophistication of the technology when you start studying your ATPL's.
Hope that somewhat answers your question.