The starting dose depends on many variables -- if you are <50yo and have no heart problems, a TSH of 7.44 usually gets a start dose of 50-75mcg.
My doctor started my out on 100 mcg with a TSH of 29. With your THS level I would assume your doctor would go low(er) and slow as not to over dose you, which could easily happen with TSH 7.44. However we are all different (and so are doctors) with different circumstance to our thyroid conditions.
Mild hypothyroidism can often be treated by a primary care physician.
I have always been treated by an Internist in my 12 years thyroid. However, it is recommended that certain types of hypothyroidism patients see an endocrinologist: Patients of age 18 years or less, Patients unresponsive to therapy, Pregnant patients, Cardiac patients, Presence of goiter, nodule, or other structural changes in the thyroid gland, Presence of other endocrine disease. Unless you fall in those categories, I would say its up to your judgment in which doctor you would prefer to be treated by.