This is how it was back in the 70s-80s: After a few jumps where your chute opens automatically because the rip cord is attached to the plane, you could sign up for training that would qualify you for more ambitious jumps which allow you to freefall and open your own chute, hopefully before you hit the ground. Initial freefall jumps are always done with the instructor, in tandem. Only experienced jumpers stretch out the freefall to where they are close to the deck. Freefall jumps are from much higher altitudes. There is no point in doing them from the low altitudes that static line jumps are made from. It's obviouly dangerous, and the whole idea, the enjoyable part, is the falling, the flying, the control, the ability to alter direction and attitude, the spectacular feeling and vision. This is mostly self- taught as you do additional jumps. There really is no other way. As you get better and more experienced you can try new things, hold off popping the canopy until you are very close to the ground, watching your altimeter unwind like a maniacal clock. There are also specialized manuevrable parachutes, almost like gliders. I have no experience with them at all.
The answer to the seond part of your question regarding instructors encouraging you to pursue further training, the answer is mostly no. Becoming an instructor? Absolutely not. It's not that kind of sport. There is a great deal involved in running jumps. Because sloppy or inept prep and procedure leads to instant death there is a great seriousness to the whole thing. There is a certain solitary quality inherent in sky diving. Very very few casual jumpers opt for the advanced training that qualifies you for freefalling jumps from almost 3 miles up. Only a few first time jumpers do even a second jump, and not most that do only do a very few. A lot depends on what part of the country you are talking about. There are strong regional elements and differences.