I am 37 years post total thyroidectomy for cancer, so I have been on medication for all these years. I like to think I know a thing or 2, but you may be getting plenty of advice already.
I only want to say that you should be certain you are getting not only the TSH, but also a Free T4 AND, very importantly, a Free T3 test. Doctors so rarely do that FT3 part and it is actually very important. I won't go into explanations, it may be more helpful to "do the google," as George W would say. It has taken me way too many years to understand the importance of that test.
Yes, I was on Synthroid for most of those 37 years, but I am now on Armour (a natural, not synthetic drug,) plus I take a T3 supplement called Cynomel (because it comes from Mexico) and I can't afford the American version called Cytomel. You have to work your way through this.
When I had my cancer 37 years ago there wasn't an internet or books. Now you can go to the library or a bookstore or Amazon and there are almost too many books. If I had those available to me way back then my life would probably have turned out differently. Thyroid problems can really mess you up. Arm yourself with information before you ever talk to your doctor. Also, depending on the outcome of your tests you should probably ask about antibody tests to look for things like Hashimoto disease. Don't know what kind of doctor you are working with, but it is probably best to start out with an expert in all things thyroid. Basic internists don't always keep up with these things.
Most importantly, never forget to ask for a hardcopy of your test results. Another thing I learned the hard way. Then, get some of those books on thyroid info and look up test result info. Know what the results are telling you and don't ever rely on your doctor to do that.
Don't mean to be preachy. I learned these things the hard way. You don't have to. Look at the books now or start online and you can easily get symptom information.
Best of luck.