Excellent goolarra, thank you.
I think the really important thing to do is pick a lab and stick with it...that eliminates one variable. Although all labs make mistakes, I've never heard of one that is consistently bad.
"Antithyroid microsomal antibodies" is another name for TPOab. "Thyroglobulin antibodies" is TGab.
TRab is another antibody test, usually ordered when Grave's (hyper) is suspected. There is a better test for Grave's, but since your symptoms are hypo, I'd skip this one. I don't know much about the first two, except that I believe they're kind of obsolete (check me on that!).
Really, the basics for diagnosis are FT3, FT4 and TSH and the antibodies, TGab and TPOab (if you suspect hypo). These tests will give you a lot of information about your thyroid function, will suggest whether or not you should be on meds, and will give you a direction for further testing (if necessary). Further down the line, other tests might be indicated, but these are a very good start.