*WARNING* The first couple of paragraphs are pretty basic but the rest gets pretty in depth and it may not all be as coherent as I would like it to be. I apologize for being so lengthy but I've been dealing with this for as long as I can remember so it's just sort of second nature to me. I'll separate the simple from the in-depth with a row of *** - Let me know if anything is wrong or confusing and especially if it helps! Now, here we go:
I would definitely suggest having them test your Free T3 and your Free T4. These are not absolute indicators of a problem but they do seem to be more accurate when dealing with a borderline problem or a problem that is masked by other issues.
When dealing with thyroid issues what we really need to know is the energy output of an individual cell (or lack of) but since that's impossible we try to find out what the body is doing in reaction to that output. Your T3 and T4 levels are the levels of hormone that are floating around in the body whereas the Free T3 and Free T4 are the levels of hormone that are available for use by the body. This is very important because a lot of these hormones will bind to other elements in the body and a TSH level can read normal, along with a normal T3 and/or T4 because the body is technically producing enough and there is technically enough in the body but it is not all useable.
Your TSH is definitely an important number to have because it tells you what the pituitary gland is asking for; speed it up or slow it down (production of thyroid hormone, that is). TSH is Thyroid Stimulating Hormone which confusingly enough means that the results are read "backwards", in a sense. If your TSH is high it means that the pituitary gland isn't finding enough of the thyroid hormone(s) in your blood and it is asking for it to produce more (ie. hypothyroid) whereas if your TSH is low the pituitary gland is finding too much hormone and is telling the thyroid to slow down (ie. hyperthyroid).
Just to complicate things further, there are actually other processes going on with the T3 and T4 (these are only 2 of the many types of hormone that the thyroid produces, all of which are important in varying degrees and for different reasons). Both of these hormones are produced directly by the thyroid but not both in the amounts that are needed. The body uses what it needs of the original form but then it converts T4 into (a slightly different but necessary form of) T3 which it then uses for various functions. (caveat: these hormones also convert down into T2 and other T's that are important for things such as metabolizing food/sugars) There can be a normal output that is thwarted by a problem with conversion.
To further confuse things, we have absorption. There can be plenty of hormone that is converting properly and available for use in the proper amounts but for some reason the cells can't absorp it in the right quantities. This can be caused by something as simple as a vitamin or mineral deficiency (too much calcium) or as complicated as "thyroid resistance syndrome" (can be caused by an unbalanced ration of omega 3/omega 6 fatty acids).
Other medical issues can look like a thyroid issue because they directly affect the levels of thyroid hormone in the blood. For example, an elevated cortisol (adrenal hormone) level can alter your thyroid test results either making the thyroid issues worse or making it look like there is a thyroid problem when there may not be.
My best advice would be to take control of your health by being as educated as possible and by insisting on proactive medical treatment. If a doctor tells you something that sounds contradictory to what you've read or heard elsewhere then you should feel confident in asking them to explain and quantify what they're telling you. You should never feel ashamed for not understanding the technicalities of your condition. As well, you should never accept a condecending attitude from a doctor because they are human and they don't know everything. In fact, sometimes they know things that they refuse to believe are true so never let them tell you that you (as an informed patient) are wrong and they are right without giving you concrete evidence for their beliefs. Finally, if you feel neglected or insulted and you've questioned (doctors are overworked in many instances and misunderstandings can occur so you have to be understanding and give them an opportunity to explain before assuming that they are "one of the bad ones") and been either misinformed or ignored (this includes refusing to release or explain lab results or choices in treatment as well as withholding results or files) then you should leave and find another health care professional immediately. You are entitled to know everything about your treatment.
Sorry for the militant tone but I believe that society as a whole has elevated doctors to an irreproachable status and it's time we take back responsibility for our health and insist on a partnership with those who have chosen a profession that is suppose to help and heal. We have to force those who are unwilling to recognize our part in all of this to listen to us regardless of whether it makes us seem difficult or not. It's not about being pushy or rude; it's about taking responsibility and holding all of those involved accountable.
I hope this helped! I look forward to getting to know everyone around here!