Generally speaking any modern jet engine is started by mechanically driving the inner spool of the engine (there may be up to three on RR-Engines). This is done via a bevel gear set, a shaft leading to the outer side of the engine and a gearbox attached to it. It depends what drives the gearbox, it might be an electrical starter (mostly found on small business jets), a combined starter and generator (rare), an hydraulic starter (also rare) or a pneumatic starter (widely used).
These starters use really high rpm, so the effectivity of electric starters is low (high speeds on the brushes, high inductive losses), same is for hydraulic starters (high flow velocity of the hydraulic fluid) pneumatic starters take much more rpm without getting uneconomically.
If inner spool rpm is high enough (normally in the order of 30%) the combustion chamber pressure is high enough to make it work (vaporizing and burning the fuel), the outer spool(s) is(are) driven by airflow via the according turbine.
I once cranked a small gas turbine (military water pump) by hand ! Very interesting thing, you can feel the torque needed to drive the engine, feel it increase with rpm (more air blown through), feel it decreasing if fuel and ignition is switched on, feel it further decreasing the faster you drive it until it runs on its own at roundabout 40000 rpm. (you have to be FAST, because turbine temperature rapidly rises if you ignite it, and stabilizes only above 50000 rpm !)
Very interesting experience !