Forgot to mention that hypo patients are frequently low in other areas as well. So I suggest that you also get tested for Vitamin D, B12, ferritin, and a full iron test panel.
Even more important than the test results would be how you are feeling. If you look at this list of 26 typical hypo symptoms, which ones do you have if any?
http://endocrine-system.emedtv.com/hypothyroidism/hypothyroidism-symptoms-and-signs.html
In order to best assess your test results, we really need to know the reference ranges for those tests, as shown on the lab report. The reason is that test results and reference ranges vary from one lab to another, and it is important to know where the result falls within the range. If you will lpost the reference ranges it will be very helpful to us.
In the interim, I will speculate based on reference ranges that we frequently see used by other labs. From that I would say that your T4medication has raised your T4 level adequately, but your T3 level has not changed. Note that these tests appear to be for Total T3 and Total T4. Those are not nearly as valuable as tests for Free T3 and Free T4, so it would be best to request Free T3 and Free T4 in the future.
Since your T3 level did not increase as your T4 level did, that is an indication of inadequate conversion of the T4 in your med, to T3. Total T3 reflects the total amount of T3 in your body. Most of that is bound to protein molecules and thereby rendered temporarily inactive. Only the small, unbound (free) portion is biologically active. Free T3 largely regulates metabolism and many other body functions. Scientific studies have shown that Free T3 correlated best with having hypo symptoms, while Free T4 and TSH did not correlate. Many of our members, myself included report that symptom relief for them required that Free T3 was adjusted into the upper third of its range and Free T4 adjusted to around the middle of its range.
A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms. Symptom relief should be all important, not just test results. You can gain some good insight into clinical treatment from this letter written by a good thyroid doctor for patients that he sometimes consults with from a distance. The letter is sent to the Primary Doctor to help guide treatment.
http://hormonerestoration.com/files/ThyroidPMD.pdf
I take it as a positive sign that your doctor plans to test for Free T3, but TSH is almost meaningless after starting on thyroid medication. I would even suggest that you give your doctor a copy of the above letter and ask if he is willing to treat clinically. If not then you are going to have to work on him to change his mind, or find a new doctor that will do so.