Hey - nice name! The fish, I mean.
I've owned bettas for a total of.. three weeks. Bought fish, did homework, freaked out, fixed it. The baby analogy is very apt, I think.
My daughter's HM had a similar pinhole in his dorsal fin when he got home, and a very slight tattering in other places. It doesn't look like rot, but I've taken the precaution of salting his water (there's a lot of posts here on how to use aquarium salt properly for gallons, I used the litre-centric pack directions). It helps to heal fins and other problems if used for a short time.
My own betta Sid was a red lump of skinny mope for two weeks, until he settled into his 3.5g tank. He refused his flake food for days. I fed him some pellets and live food, he perked up immediately, and steadily got happier. Then he got cotton fin disease. That went away in a matter of two days on its own accord, presumably clean water being a factor. Then he got ich.
Sooner or later, the crash course torrent of information catches up with your fish stabilising in his new environment and you can finally relax a bit.