Should I get any additional tests other than a TSH to know if I'm hypothyroid?

Maybe. A TSH is an excellent screening test, and most hypothyroidism will show a high tsh. Pituitary problems can cause hypothyroidism with a low or normal tsh. There are some other medical conditions that can supress the tsh. A free T4 is an easy way to confirm the thyroid status. Keep in mind that many of the symptoms of hypothyroid (fatigue, dry skin, etc) are caused by a lot of other conditions.
Possibly. For the vast majority of people, TSH (thyroid stimulating hormone) is the best test to assess thyroid status because it measures one's own response to the levels of thyroid hormone in their blood. However, for someone with certain brain or pituitary problems, the pituitary may be unable to make appropriate levels of tsh. For these people, measuring T4 with the TSH is important.