All correct so far...here is what I would suggest. As said, if you have any aspirations of an aviation job (i.e. commercial certificate or better...), you're going to want to know if you can receive the required medical certificate. However, you're also going to need a Student Pilot Certificate, which just so happens to be on the back of a Third Class medical certificate. So, if you're going for training, I would go to the AME, get a THIRD CLASS medical, but ask him to tell you if you would be able to pass a FIRST CLASS. There are some extra vision requirements for second and first, but other than that, not too much different. The reason you wouldn't want to just GET
the second or first is because then you'd need to go to a Designated Pilot Examiner and get a Student Pilot Certificate, for which they would charge you another $25 or so that you could have avoided. Expect $80-90 for the medical. As also posted before, the three classes of medicals usually cost the same because they are so similar, so the doc shouldn't have any problem with this.
Enjoy your training!!