Let me add to what's been said that 1400' grass strips at low elevations are pretty doable for experienced pilots, but for newbies, it's a lot to ask, especially after flying all day to get home. At the other end, you'll be dealing with high elevations and high density altitudes. You didn't say where in NM, but all of NM is pretty high country. As good as 150s and Tripacers might be, they're a lot less usable in the high country in warm weather--or as we used to joke in Laramie, "they taxi fast in the summer".
On commuting between MS and NM, I agree that your biggest issues will be weather related. 800 miles in a 172 is a full day--I do it every year going to OSH and back. Although I have done it totally VFR successfully, it's a lot easier on an IFR flight plan, because often enough, I've run into IMC coming or going or both. In past years when my kids were small, we took our annual vacations via 182, and our 1100 mile trips every year for several years were VFR only once, going. We'd have been there an extra week if we couldn't have come back IFR. In those lengthy cross countries, it's not unusual to run into 3 or 4 weather systems, so planning becomes more difficult, with the real possibility of having to wait out weather overnight. Although in 42 years of flying, I've only had to wait overnight a handful of times, I've certainly had to delay leaving.
The fact is, no matter how we like to deny it, going places by small airplane is always problematic, usually because of weather, but often enough due to some mechanical issue. The old saw, "time to spare, go by air", is realistic.
On buying an airplane to learn in, while many people do successfully purchase an airplane to learn in, my own view is that renting is better, rather than beating your own airplane up as a student. While it may seem that the rental money is going down the drain, you won't have maintenance to contend with--maintenance and airplane ownership in general is expensive, plus hangar/tie down and insurance. Unless you go the experimental route, which will usually save some, you're stuck with high priced parts and professional maintenance--there is only a relatively small list of what a non-aircraft mechanic can do to a certificated airplane. If you do go experimental, and unless you build the airplane yourself, you'll still be paying a professional aircraft mechanic to maintain it, although there are more things you can do yourself, and some pieces and parts are less expensive. The bottom line here is that you really don't save a whole lot by owning rather than renting--and you have to fly a lot of hours per year to break even, due to the fixed expenses.
I don't disagree at all with the idea of a Mooney for traveling--much quicker--but I wouldn't recommend one as a trainer. They're not difficult to fly and frankly a lot of fun, but they're a whole lot less forgiving than 172s or PA28/160s.
Lots to think about, huh?
Cary