Hi Vicki and welcome to the forum.
You have asked a few questions:
1. Symptoms - it takes a while, and probably several dose increases for symptoms to ease typically. To avoid shocking the body, dose increases should be a maximum of 25mcg a time, no closer than 6 weeks apart. This process should continue until tsh is around 1, then move on to fine tuning. Which leads on to
2. Levels - within normal ranges only really applies to healthy people not on replacement thyroid hormone. Once diagnosed as hypothyroid and on levo, we generally feel better when tsh is around 1 or below, with ft4 in top quarter of range and ft in top third. We are all individuals with different responses and requirements, hence why I mentioned fine tuning above. Unfortunately many doctors appear to believe that "in range" is good enough - not so!
3. Time- as in 1 above, you are likely to need several increases and yes it could all take a while. Nothing changes quickly with the thyroid unfortunately. It may also take several months for relief of symptoms even after all blood levels are optimal.
4. Chemotherapy - I had chemo a year before finally being diagnosed with hypo. I am not sure that it has a direct effect on the thyroid, however, it does damage the gut. Especially with the auto-immune form, this can weaken the lining enough to allow a real ingress of foreign particles (main offender gluten) into the bloodstream, irritating the immune system. This can either dramatically increase the scale of an underlying existing problem, or be the initial trigger for an underlying genetic pre-disposition to auto-immune. Chemo can also leave the immune system reeling - I had severe neutopaenia, for instance, during treatment, effectively an immune system wipe-out - that can't be good and has to have the potential for a slightly skew-whiff immune system. Read up on leaky gut, especially if you have any digestive issues.
On general points, a good place to learn the basics is our parent website thyroiduk.org.
It is almost universal to have serious nutritional deficiencies with hypo, caused by low stomach acid and malabsorption due to gut issues. Fairly high (not just in range) levels of vit d, vit b12, folate and ferritin are required for thyroid hormone to be used effectively. it is also important to know if you are anti-body positive for Hashimotos auto-immune. If your GP won't test all of these, many of us use private testing through Medichecks (thyroid ultravit) or Blue Horizons (thyroid plus 11).