About the age of the airplane - I’ve flown airplanes 3 months old, and 60 years old. Provided they are properly maintained, both are good. Don’t be afraid of an airplane 20, 30, 40, etc. years old (again, if properly maintained - they are supposed to get a yearly inspection, and an inspection after every 100 hours of operation, and you can ask to see the logbooks. A reputable operation won’t hesitate to let you see them.)
I found a 1962 Cessna 172 today… for $8,000 (Barron Thomas site). Ooh, that’s tempting! That’s the ‘razorback’ model with the swept tail. 
But A) I’m not ready to buy a plane, and B) There were no engine logs.
There are ‘mom & pop’ FBOs, like the ones I trained at, ‘small business’ FBOs, like where my mom worked, and ‘Big Corporation’ FBOs that advertise in flying magazines.
‘Mom & pop’ FBOs tend to be small, local operations that are not necessarily (or even usually) run by ‘mom and pop’. I learned to fly at Barnes Aviation, which is a family business started by Pancho Barnes’s son Bill and his wife. After his fatal crash his wife Shuling took over the business. The last time I contacted Barnes Aviation, it was being run by a son. Group 3 Aviation was bought from Santa Monica Heli Services by three instructors. They are not related, but I think it counts as a ‘mom & pop’ type of operation.
Gibbs started out pretty small – I mentioned the one-room shack – but grew considerably even in the mid-'70s when I moved away from San Diego. I call that sort of FBO a ‘small business’ type, since they are larger than ‘mom & pop’ ones and similar in structure to many small businesses.
I’ve never flown at a ‘Big Corporation’ FBO. Maybe United Beechcraft at VNY qualifies, as part of its operations catered to business aircraft. (I used to see Beech Starships parked there.) Mostly my impressions come from ads I’ve seen in the magazines. For example, there’s a heli operation in Concord, CA that touts is massive fleet of Schweizers, and there are similar ones for fixed-wing. The ones I call BC FBOs are the ones that advertise courses from Private through ATP for a fixed price, or have other ‘packages’ for fixed prices. My impressions could be wrong, but they are what they are.
I would steer clear of ‘Earn your license in x days!’ and ‘Earn your license for $x!’ shops. They are required to provide sufficient training for an applicant to meet the requirements to pass the Private Pilot (and other) check ride. If they send too many students to the Designated Examiner who are not prepared, the FAA is going to give them the stink-eye. So I’m sure their training is at least adequate. But where I trained, the emphasis was on quality. This was specifically stated at Group 3, and pretty evident at Barnes. The goal of my instructors was to make their students safe and competent pilots. If a student was ready for his check ride in the minimum 40 hours (I was ready at 38 when I got my first (fixed-wing) rating, and my instructor sent me on an extra cross-country to meet the time requirement) or 100 hours. My instructors would not recommend a student for a check ride until they felt the student was prepared to be set free with a little grey piece of paper (which licenses were at the time).
Now, a cynical person might say that ‘making sure a student is prepared’ is code for ‘Let’s milk as much money out of this sucker as we can.’ Maybe that’s true in some cases; but I felt that my instructors really cared. A dedicated flight instructor gets a sense of pride when s/he solos a student, and again when the student officially earns his wings. That’s the kind of instructor to look for. Two of my instructors were basically retired, and were doing it for the love of it. (And they make a few bucks.) The other two were my age (young, at the time) and wanted to make money. One went on to fly helis for a corporation, and the other is still running Group 3 (very successfully, last time I checked). All were dedicated to turning out safe and competent pilots. That’s the kind of instructor to look for.
As mentioned by others, choose an instructor you get along with. Personalities sometimes clash, and sometimes they mesh. There are enough personalities out there that any prospective student can find one to meet his or her needs.
As Broomstick said, most pilots have been trained in Cessnas and Pipers. When I a kid, you basiccally had a choice of a Cessna 150/152 or a Piper PA-28 Cherokee. Nowadays, from casual observation, the popular planes are the Cessna 172 Skyhawk and the Cirrus SR22. The Cessnas seem to be used more by smaller or older operations, and the Cirruses seem to be used by larger or newer operations. These observations are from this area and are not meant to be a global declaration. You can train in just about any airplane. Mostly though, you’ll see Cessnas, Pipers, and Ciruses.
I trained in dad’s 172. I like high wings because the downward visibility is better than in a low-wing. You also don’t have to get on the ground to drain the sumps. And there’s shade from the sun or shelter from the rain. Some people like low wings because they have better upward visibility. Maybe some people think they look better than a prosaic ol’ Cessna. My low-wing experience is five hours in a Grumman AA-5B and a couple/few hours in a Piper PA-28-235 Dakota. I liked the sporty Grumman with its sliding canopy. The Piper was nice enough, but I don’t care for a single door and having to slide across to the pilot’s seat.
Agree with the folks above who say it doesn’t much matter which airplane type you choose to learn in. But not mentioned yet: Once you start in one type, it’s quickest & cheapest to complete your training in that type. For that reason there’s some benefit to picking a school which has more than one airplane of the type. That way you’re not stuck if it’s out for maintenance for a few days.
It’s definitely good to fly a bit in different types as you build your experience. But when you’re just starting out through, say, initial solo, trying to learn in multiple types will just add confusion & difficulty.
Another option to look at is a flying club. Typically you buy a share in the club and then get to rent the planes for a cheaper rate than most FBOs. You will also probably pay montly dues whether you fly or not, but if you are training you will be flyinmg quite a bit
shameless plug: http://redbaronflyers.org/
Brian