Sean,
At the end of your training you should have a CPL + IR with multi engine and most likely, all your ATPL subjects done.
You'll have approx 200 hours or so.
To be a captain of a Lear will require at least 1300 hours more at the absolute minimum. Why? You'll need to hold an ATPL. That's a minimum of 1500 hours (with specific hours required for night etc). Most operators stipulate a minimum of 3,000 hours.
Lear jets are above 5700 kg, so an ATPL is required.
Air taxi is pretty much as it says on the tin. It's ad hoc charter. It can be in any thing from a Chieftain (light piston twin- I think they weigh about 3,000 kg) to a Gulfstream G550 (41 tonnes, range 14 hours). (Although, that is a tad overkill)
Charter can be done in airliners. In fact, a lot of it is done like that in Europe.
Charter can simply be a regular flight that is not open to the public, eg oil rig transfers.
GA is challenging. It does encompass a wide variety of flying. My husband flies for easyJet, he'd swap his job for mine in a heart beat.