becoming a DO is pretty much the same as becoming an MD. There are schools that only train professionals to be DOs, just as there are schools that only award MD degrees. As far as I know, you can't go to just any medical school and then choose to become one or the other, you have to go to a school specifically for that.
Similary, on the dental side, there are two degrees, DDS and DMD. They are identical, unlike MD and DO, but you can't go to a school that gives DDS degrees and ask to be given a DMD, or vice versa.
That is essentially correct. There are two degrees in the United States that allow someone to practice the full scope of medicine, as a physician, an MD and a DO. The two degrees are professional and legal equivalents, and physicians who have either degree practice to the exact same standard of care when treating patients. The main difference is that DO's must learn OMT during medical school, which is something you'll want to investigate and learn more about if you are considering attending a osteopathic medical school. The two types of physicians work side-by-side and often train in the same residencies/fellowships, since a majority of DO's now opt to match into an ACGME residency, instead of their native AOA residence, for a variety of reasons.
As lemoncurry said, to become a DO, you must attend an osteopathic medical school. MD's attend allopathic medical schools. In general, the two types of medical school are separate entities, accredited and governed by different organizations (AOA/COCA for osteopathic medical schools and AMA/LCME for allopathic medical schools), but MSU actually has both programs. There are many more allopathic medical schools in the US than osteopathic ones, just as there are many more MD than DO physicians. The general pre-requisites for application to both types of schools are essentially identical. You can proceed with the usual premed stuff, whether you are applying to MD or DO medical schools, but if you think you might be interested in becoming an osteopathic physician, it would be a good idea to learn a little about the history of the tradition and shadow a practicing DO; getting a letter of recommendation from a DO would be a good idea, since some osteopathic medical school require it and many prefer it. When it gets close to the time you want to apply, whether MD or DO, you'll want to check specific requirements by school, since schools may have variant or individual requirements for application.
For the most part, the curriculum for all four years of medical school is nearly identical between osteopathic and allopathic medical schools, with minor variances and differences in emphasis, some more related to the mission of an individual school than by degree. You won't be able to much tell the difference. I am an osteopathic medical student, currently in my second year, and we do pretty much the exact same things as our allopathic neighbors. However, as I mentioned earlier, one of the notable differences is that you'll have to learn OMT and it is integrated into the curriculum during the first two years of medical school (and in some cases during your rotations, too). Another major difference is that DO's are required to take the COMLEX series of board exams to become licensed. It is similar to the USMLE, but may have differences in question style, curricular emphasis, and, of course, the addition of osteopathic principles and practices. Osteopathic medical students have their own residencies (AOA), but can also match into allopathic (ACGME), or dually-accredited (AOA/ACGME) ones, if they choose. If you are attending an osteopathic medical school, but know you want to apply to an ACGME residency, it is a good idea to take the USMLE Step I, in addition to COMLEX, so that program directors can compare apples to apples, although most ACGME residences do accept the COMLEX.
Check out the AACOM, for more information on the osteopathic medical schools, a bit about the osteopathic tradition, and how to make application:
http://www.aacom.org/InfoFor/applicants/Pages/default.aspx
Good luck!
