First of all, no need to be embarrassed. We all had to start somewhere, and some of us (like myself) who started young didn't necessarily start out "right." Lots of what I had learned early had to be unlearned, and I had to learn the "right way" on some things the hard way, which was in adulthood. Ask questions, and learn, and this is a good place to do that. Some answers about tactics will be area or situationally correct, but may or may not work for you in your area or situation. Whatever you do, put different tactics into practice, find out what works, always keep learning, and start a hunting heritage for your kids.
I think that rifle selection is largely personal. Opinions on rifles are like belly buttons and buttholes. Nothing wrong with any rifle on your list. I would personally save a little more and go with a Tikka or Bergera if I could, or I'd try to find a used Savage 10/110/16/116, etc, BUT you know your preferences, budget, and timeline better than I do. Starting with something less expensive and later upgrading is not a bad thing, especially for an adult onset hunter. You can find your likes and dislikes which could influence your later decision on something a little bit nicer (if you desire to do so later), and if you don't like what you buy then you can always sell it and not take much of a hit.
All of that being said on cartridge… I won't dive into ballistics and specifics on terminal performance. I'll stick with the two that you listed, and EITHER ONE WILL WORK JUST FINE. I'm not a huge fan of the .243, but I won't deny that it is effective. Recoil is light, and it is sufficiently powerful for hunting so long as you use quality ammo AND are selective with your shots and stay off of the shoulder. But the .308 definitely has more horsepower for the angled and shoulder shots, and it will provide better frontal diameter and a greater chance of a more consistent blood trail. Ammo has also traditionally been far more available and affordable for 308, so that's another win. Also, recoil is not bad for 308 with 110-150gr bullets, but you have a lot of flexibility for more bullet mass with the heavies as well. If recoil is bothering you then find some lighter weight cheap ammo to practice with, then do your final sight in with a quality 130-168gr hunting cartridge for hunting.
MANY other cartridges will work, including 6.5 creedmoor, 7-08, 270 Win (my favorite), and so on, but that can easily lead us down rabbit trails.
