I started to feel more human after just a couple of weeks of taking thyroxine and probably back to normal after 6 months or so (difficult to say, as mine failed just after having dc1, so my perception of how I should feel was a bit skewed due to having a whole new person to look after!).
My dosage was adjusted a few times (I felt ok but blood tests showed I was a bit overactive taking 150, so went down to 125 for a couple of years. Weight has been up and down a bit, but when I eat less and do more I don't have any trouble losing weight.
I now take 100 most days, add in 25 when I feel a bit sluggish and for the most part I feel perfectly normal (well, as much as any single mum of 3 DCs can feel normal!)
I recently diagnosed myself with B12 deficiency, which felt quite similar but with added numbness in my extremities (face, limbs and 'other areas'!) but with a course of injections the numbness was thankfully reversed before it caused any permanent nerve damage and now I take some vitamin supplements as a precaution rather than rely on my GP to order tests and recommend more injections.
It's wise to read up on the other things that can occur when you have an auto immune condition like Hashimotos. The B12 thing has always been at the back of my mind as a possibility, i don't know why, so when it happened I was almost prepared for it. My GP was quite dismissive when I mentioned it, but did the test anyway and she was annoyed surprised when it came back as low! She would never have thought to test for it, given my symptoms - she wanted to send me to a gynae specialist for the numbness down there, so it is a good idea to educate yourself a little, try not to scare yourself with the possibilities but be aware.
There may still be other little niggles sometimes, I don't deal well with stress and get very tired if I overdo things, but on the whole, I don't think I suffer with tiredness any more than most busy mums. I'm lucky to work pt and have a wonderful supportive dp who takes good care of me and brilliant DCs who are understanding when I'm a bit pooped and just want to lounge on the sofa for a while, but there isn't really anything to show that I have this condition, other than the little tablet every morning.