Synthroid is for hypothyroidism. Dose adjustments are common in order to optimize replacement - TSH levels are followed in order to help determine the best replacement.
Taking synthroid means you're hypo. Usually you start at a low dose and increment it up till the TSH is down to a normal level ( the normal level of TSH varies a bit depending on who you talk to but its at least below 4 ) which means that you should have enough thyroid hormone in your blood. In another 6 weeks they'll want to run another blood test of your TSH ( and probably FT4 ) to see if its normalized or whether you still need another increase of the synthroid. You should be able to get your labs from your doctor and if you want you can post them here for further advice.