The cheapest way to get training and build time? Absolutely no question - sign up and go to work for Uncle Sam. Good training, good equipment and he'll let low-time inexperienced guys fly everything that he has. Oh, there's just one catch...
Buying an airplane is a very viable option, but there are others - join a "not for profit" flying club or take partners in on something. Just make sure your bases are covered legally. (Partnership agreements, insurance, etc.) Personally, back when I was working on my commercial and instrument ratings I bought an airplane with a couple of other guys. When I finished my training, we sold the airplane and made enough to cover the costs of our training and I actually had a little money left over.
It absolutely makes no difference what type of airplane you learn to fly in - high wing, low wing, Piper, Cessna, whatever. However, there are some real benefits of having some "tailwheel time". They make for some very demanding training, but you'll be the better pilot for it. The problem is finding a COMPETENT taildragger instructor. Now days, they are getting very hard to find. Not just any CFI will hack it. If you go the taildragger route, you'll most likely have to budget some money for some time in something with a full instrument panel for your checkride and a few hours of training. However, you'll still be able to do most of your training in it.
If you decide to buy an airplane do it based on ease of resale. You're not going to own it forever, so look for something nice. You'll probably want to avoid that ratty 150 out on the back tiedowns. Just find a good airplane that you'll be able to easily resale, but make sure that you (and your partners) can handle the maintenance, payments and insurance. Stack the odds in your favor - look for something with a fresh annual from a reputable shop and a low to mid-time engine. Hopefully you'll be able to jump in, do your training and jump out before any serious work needs to be done. With a bit of research and a bit of luck you might be able to break even or better. I've owned several airplanes (and gliders) and I've never lost money on any of them.
When it comes to building experience, I believe that flight instructing is a necessary "evil". Actually, you can make of it whatever you want. Personally, I find it to be (for the most part) an enjoyable endeavor.
I would recommend that you get your CFI and spend some "quality time" teaching. Remember, in any teaching situation, it's always the teacher who learns the most. The same thing applies in aviation. Getting your CFI and actively instructing for a while will teach you things about flying that you will only learn through instructing. In my case, I had an ATP and over 3,000 hours before I got my CFI. I thought that somehow I had "dodged a bullet" by being able to go directly into Part 135 (charter) flying after I got my commercial and instrument rating. I was wrong. I ended up getting my CFI certificate so that I could keep my hand in flying while I went to school full time to finish up my degree. I learned a lot and to this day the lessons are very valuable and useful. Don't avoid becoming a CFI, in my opinion it is a valuable step in your aviation career.
The real question is how long do you need to be an active CFI? My personal opinion is that you'll probably get about all your going to get out of flight instructing after 500 to 1000 hours. Remember, there's a big difference between 2,000 hours of experience and 1 hour of experience repeated 2,000 times. Flight instruction tends to fall into the latter category.
Oh well, you guys can probably tell that I've got entirely too much time on my hands - the airplane is in the shop this week for maintenance.
Lead Sled