Hi Slippery
No question is too dumb here so don't be shy!
I initially trained on General Aviation aircraft on the Civil Aviation Register (VH registered, things like the Piper). If you like you can train straight away in GA, but it is more expensive. Many people now train in RA-Aus aircraft (like the Jabiru) then convert to flying GA aircraft like the Piper. It's cheaper to do that than to train on the Piper from scratch, but you can choose which way you want to go - maybe talk to your instructor. Ian has mentioned the hours required.
If you pass the GFPT (General Flying Progress Test) you can fly aircraft like the Piper and take passengers, but you can't navigate far from your home airport. To be able to travel in the Piper you will need your PPL(Private Pilot Licence), which includes navigation training. The "navs" can be a bit expensive as you might be doing 3 hours in a flight. Your school could advise the rate per hour for this.
There's quite a lot of theory to do. The theory tests for the PPL do take some study but there are good books available, courses you can do, and your instructor should give you some guidance. You'll have to learn about things like aerodynamics, aircraft systems, and the rules of the air. When you start navigating you'll have to learn more about weather and navigation. You'll start learning some basic theory in your lessons right from the beginning and you'll build on that as you go.
To fly in cloud you will need either a private instrument rating (known as a PIFR), or a command instrument rating. The private one can be done in stages, so you only learn as much as you need, and you can do a theory test before you move to each new stage. If you want to go the whole hog you'll need a command instrument rating, that's 40 extra hours of flying (although some can be done in an approved simulator) plus a theory exam called the IREX, much of which is about the regulations and procedures for instrument flight. The IREX has a bad reputation because a lot of people do fail, but with study it is not too bad. Remember, you aren't going to sit the IREX now, by the time you get around to it you would have a lot more knowledge. So for cost you would need to budget for at least 40 hours (plus the flight test) in a suitable aircraft. Either that or you could do a PIFR, or do what many pilots do and avoid flying in bad weather.
Your instructor should be able to run you through the details. Remember it does take some time, so most people tend to do things in stages.
You won't need to buy an aircraft if you decide to hire aircraft from the airport. That means they will pay for the insurance, hangarage, maintenance etc and you pay an hourly rate. You could then try out different types to see which you prefer.
As Ian said, it is difficult to say which aircraft is best because they have differences. Some are designed for speed, some to fly slowly, some for aerobatics, some to be easy to fly, some are more powerful and carry more weight. Some pilots prefer high wings like Jabirus or Cessnas, others prefer low wings like Pipers. As you progress hopefully you'll start to realise the type of flying you prefer. If you have specific questions about types you could ask on here.