I'm picking up guitar and would like to know generally how long I should give my fingers to strengthen and callous if I practice 15-20 minutes 5 or 6 times a week. My concern is that the guitar is possibly too big, or the strings set in such a way that they are too hard to press chords crisply, do barre chords, etc. I don't completely know if this is a legitimate concern, but I'd rather not jump to going to a shop to have the guitar checked out and fitted without giving my hands a chance to get to an appropriately expected strength. After 3 weeks, my hands still feel tired pretty quickly, and I'm willing to give it whatever reasonable time to strengthen before deciding maybe there are sizing factors that impede my playing.
Callouses should only take a couple weeks to a month but will take longer to get to their full thickness. Strength will be something you build and maintain as long as you play. You'll also learn precise finger placement that will let you fret well with less strength. It may take a few years before strength is not a concern for you at all.
If you are learning on an acoustic then you have a lot more work to do. Acoustic actions are typically slightly higher and their strings are slightly thicker than electric strings, because of the need for acoustic impedance matching at the bridge. Electrics do not need to transfer energy at the bridge and therefore can be strung with lighter strings. Proper sleep, diet, and exercise can help you build hand strength just like for any other muscles. And a teacher can help you build techniques that require less strength. If you are on an acoustic, I would look at avoiding and/or substituting barre chords, since those are the most taxing (even after 18 years of building muscles and technique, I almost never play barre chords).
Also, taking your guitar in for a setup is not "jumping" to anything. Guitars need setups periodically no matter what, maybe twice a year for a casual player to once a week or once a month for a pro (or even continuous checking and adjustment). If you've never had your guitar setup, it's usually worth it. Even if your guitar is brand new, it might just have the factory setup which is not always any good, especially for budget brands or models.
I recently took up playing again after a gap of many years. My fingers remembered what to do but the pain was pretty excruciating. I just ignored it and eventually it went away.
Little and often is the best way. Keep your guitar near you when watching TV and just pick it up during the adverts and play a few chords then put it down again. This gives your skin time to repair itself and doesn't get you to the point of massive peeling.
15-20 minutes 5 or 6 times a week is absolutely nothing. It will take you for ever that way. It's better to do 2 or 3 minutes a dozen times a day. Hence the advert technique.
Take your guitar along to the local music shop. Hang around, get to know the staff. See what other musicians come in. Ask other players (the good ones) to play your guitar and ask their opinion on it. See what it sounds like in the hands of others. You will notice shortcomings and they'll tell you about them. Unless the shopkeeper is also a really good player (or a luthier) it's probably not worth asking their opinion.
This way you will discover which problems are due to you and which are due to the instrument.
Play with other people whenever you can. This forces you to keep up and encourages you to practise harder.
At 3 weeks you are an absolute beginner so getting tired is perfectly normal. It's a gradual build up you should aim for.
Good luck and happy playing
It varies with everyone, but you should see callouses/skin peeling off after a few weeks. Also, if you get blisters, DO NOT POP THEM! Give your fingers a rest, blisters will pop by themselves when it's time.