The list of symptoms that can be possibly related to being hypothyroid is very long. This link lists just 26 of the typical symptoms.
http://endocrine-system.emedtv.com/hypothyroidism/hypothyroidism-symptoms-and-signs.html
Symptoms are the best indicator. Next best would be Free T3 and Free T4, along with TSH that they always run. Note that Free T3 and Free T4 results that are low in the range can still mean hypothyroidism, because the ranges are far too broad to be functional for everyone.
Also, since Hashiimoto's Thyroiditis is the most common cause for diagnosed hypothyroidism, you can also test for the antibodies related to Hashi's. Those tests are Thyroid Peroxidase antibodies and Thyroglobulin antibodies (TPO ab and TG ab).
Symptoms can vary depending on whether you might be hyperthyroid or hypothyroid.
With hyper, common symptoms are rapid weight loss, diarrhea, fatigue, rapid heart rate (tachycardia), palpitations, nervousness, anxiety (can also be seen with hypo), irritability, sweating/sensitive to heat, thinning skin, fine brittle hair.
Common symptoms of hypo are constipation, weight gain/inability to lose, fatigue (note that shows up on both lists), slow heart rate (bradycardia), palpitations, dry skin, hair loss, joint/muscle aches/pains, brain fog...