High TSH with low FT4 confirms primary hypothyroidism. Your GP should prescribe Levothyroxine.
The goal of Levothyroxine is to restore the patient to euthyroid status. For most patients that will be when TSH is 1.0 or lower with FT4 in the upper range. FT4 needs to be in the upper range in order that sufficient T3 is converted. Read Treatment Options in thyroiduk.org.uk/tuk/about_...
For maximum absorption Levothyroxine should be taken with water 1 hour before, or 2 hours after, food and drink, 2 hours away from other medication and supplements, and 4 hours away from calcium, iron, vitamin D supplements and oestrogen.
It takes 7-10 days for Levothyroxine to be absorbed before it starts working and it will take up to six weeks to feel the full impact of the dose. Symptoms may lag behind good biochemistry by several months.
You should have a follow up thyroid test 6-8 weeks after starting Levothyroxine to check levels. Starting dose is rarely the optimal dose you will require. Arrange an early morning and fasting (water only) blood draw when TSH is highest, and take Levothyroxine after your blood draw.