I’m an Affinity user on Windows so I can safely say it doesn’t need the internet once the software has been fully downloaded. (You’ll definitely want to open it the first time on the internet just to make sure everything has been downloaded.)
I don’t have a lot of experience with free Windows software, but I would definitely say it is worth the effort now to spend a little money now, at least the $50 one time purchase for Affinity Photo, versus going with free software like Gimp.
In the long run, having a fully featured and user friendly editor that is capable of RAW development, advanced selections, nondestructive layer-based editing, as well as advanced color correction will be worth it. You may not need the advanced features in this stage of your journey, but at least with Affinity software, you aren’t having to spend a lot of money for advanced features you aren’t using, and when you do want to use/learn them, they’ll be there.