Your largest nodule 7mm x 6mm is very small. It is unlikely surgery will be recommended unless the multi nodule goitre is compressing your windpipe or impacting nearby organs. Levothyroxine will sometimes shrink nodules and goitres but won't eradicate them. Removing nodules involves removing the thyroid lobe on which they are situated and in the case of a multi nodule goitre that usually means total thyroidectomy.
TSH 15.765 means you are hypothyroid and will need Levothyroxine replacement for life. 25mcg is the lowest dose and unless you are > 50 years of age or have heart disease it would have been more appropriate to start you on 50-75mcg.
The goal of Levothyroxine is to restore the patient to euthyroid status. For most patients that will be when TSH is 0.4 - 1.0 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, magnesium 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. Arrange an early morning and fasting (water only) blood draw when TSH is highest, and take Levothyroxine after your blood draw.