Really, your most likely path to a PPL will be through a flying school. Depending upon where in Canada you propose to learn, there will be various options open to you. The type of aircraft operated may affect your choice of school, but other than that, choosing the aircraft will not be something you have a lot of involvement in.
You will most commonly find light Cessna or Piper aircraft, with Diamond increasing in popularity on training roles. They are all fine to learn on, but each different. Ideally, you should learn on a few types, either during your flight training, or shortly after you get a PPL. There are other types, but not so common.
Alternatively, you may want to buy your own plane to learn on. This has many benefits to you, though has more up front cost.
Perhaps you may wish to refine your question, and offer a few more details about your objectives, to enable more detailed responses...
Though I can appreciate your wanting to keep the cost to a minimum, the cost is the cost, and someone has to pay it if an aircraft is to be airborne. Thus either you pay the cost, or you do something for someone else, so they are happy to pay it for you.
My early days of flying were many hours of doing "joe jobs" around the airport, in return for the odd flight. Owners of aircraft are often quite inclined to take you flying, if you have just spent the day scrubbing their aircraft to a glistening sparkle (particularly the belly). That is not as easy as it used to be though, simply because of airport security. "Keeners"' just can't hang around the ranp, as they did in my day.
Added to that, any operation flying Diamonds for training is likely only interested in dollar payment for the use of there aircraft. Some out of the way school, with older aircraft, where things are more casual offers a better chance of trading for some flying, but is by no means certain.
Your best way to economize is to pay for the flying you need to do, and prevent wasted time in the air, by being extremely well prepared for each lesson before you go, with a lot of self study, and rehersal. Though I'm not keen on computer flight simulators, they do have some merit as procedure trainers.
Remind yourself that if it is your intention to present yourself to a potential employer for a pilot job, or to an insurance company to be covered to fly someone's plane, you don't want to be presenting yourself as a person who got where they are by doing only the bare minimum.
Read "from the ground up and the Flight training manual at least 3 time, also the weather manual. Do your medical now, write the "PSTAR"? asap, wait a month then go every day for 6 weeks. If you've half a brain that should be all you'll need.
The airplane you rent won't really matter, as long as it's mechanically sound and well maintained. (They almost always look like crap.) Any new machine will cost you more, especially the new Diamonds and the "new" Cessna 172 (since manufacturing re-started.)
I'd recommend you stick with something cheap, something simple, something basic. There's no point looking for something with glass panels when you're just learning. Most schools have Cessna 152 as their normal trainers, some have 172 and some have Pipers. The only good reason to not fly the 152 is if you don't fit in it. I don't really know of any significant advantages to learning on a Piper or a Diamond.
If you want to learn tail-wheel, or glass panels, etc, do so AFTER you're licensed. It can be much cheaper that way.
As for self-studying, there's plenty of stuff online. You'll need to go through "From the Ground Up" and know almost everything in it, but you'll go through it again during ground school.
Feel free to PM me if you have any questions. (I'm in the Toronto area and I fly only for fun, which sounds like what you're looking to do since you didn't say you're looking to do a CPL.)