Why do pilots typically start out in piston-engined aircraft, rather than jets or turboprops? Essentially all pilots begin flying with propeller aircraft powered by piston engines, rather than in turboprops or light jets, even though:


Operating a piston engine requires adjusting the mixture ratio, carburetor heating, ignition timing, and other such voodoo, whereas, with a turbine engine, you basically just turn on the ignition, press the START button, and you’re in business.
Jets and turboprops are orders of magnitude more reliable than piston engines, despite the latter’s decades-long head start in development; destructive failures of aircraft piston engines are much less common nowadays than in the 1940s and 1950s, when they were practically a once-a-flight occurrence, but piston engines still come up short compared to jets, most pilots of which go their entire careers without experiencing a single engine failure of any kind whatsoever.
And when a turbine engine does fail, it does so loudly and unmistakeably - none of those deadly invisible engine failures that killed off the market for piston-engined twinmotors, along with many thousands of their pilots.
In all but the coldest climates, jets and turboprops use Jet A (in the United States) or Jet A-1 (the rest of the world), two very similar kerosenes that are far safer to handle than the avgas burned in most piston engines (Jet A and A-1 have much lower vapour pressures, much higher flashpoints, and much slower flame spread rates than avgas, and, unlike avgas, are not full of tetraethyllead), and more widely available to boot; even in those extremely cold climates where the more enthusiastically flammable Jet B (a gasoline-kerosene blend) has to be used instead, it’s still safer than avgas.
Also, a turbine engine will run happily on avgas if the ground crew refuels it from the wrong container (jets are rightly famous for running on anything that flows and burns, as long as it doesn’t produce an exhaust containing refractory oxide particles), whereas accidentally fuelling a piston engine with jet fuel will cause it to blow its insides to bits just long enough after takeoff for you to now be over a residential subdivision or something similarly inconvenient for a forced landing.
Propellers come with their own complications; the propwash coming off the propeller and hitting the wings and tail creates nasty asymmetrical rolling and yawing tendencies, the pitch of most propellers has to be carefully managed (at least during ground operations, and sometimes in flight as well), and an even slightly nose-low landing, a gust from the wrong direction at the wrong moment, or a bump or pothole in the surface can easily cause an engine-destroying propstrike.


Given all this, why do pilots generally make their debut in piston-propeller aircraft, instead of in light jets or turboprops?
In a word, money.  Have you priced time in a turboprop or light jet, vs time in a C152 or other piston trainer?  Or actually buying one?  You can pick up a decent used single for about the cost of a new car, while turboprop or jet is likely to start at half a million, and go up from there.  And add maintenance costs to that.

Then there are a lot of pilots (I'm one) whose flying just doesn't include what a jet or turboprop is likely to be good at.  I like to fly into short strips, often grass or dirt (and sometimes no strip at all e.g. dry lake beds).  Is your jet or turboprop well suited for that?

Even for someone looking towards a career with the airlines, you can build a lot of the necessary time a lot cheaper in a single-engined prop plane.  For instance, one of my flight instructors started doing that, getting paid plus lots of experience dealing with the mistakes of student pilots.  Then he moved to a job flying a small cargo plane doing pickups for FedEx or someone like that, before getting enough hours to look at an airline job.