Sorry but bettafix isn't great.. it's not a proper medication but an antiseptic that can harm your fish's breathing organ if not used correctly.. it can help clear some true fungal disorders due to the eucalypt oil in it, but that's precisely the risky ingredient for bettas.. up to you, but I won't touch the stuff.
In my 1st aid kit is:
Antibiotic (Kanaplex is the US eqivalent)
Aquarium salt (do keep this on hand! It's so useful!)
Pure epsom salt (for swelling/bloating issues)
I must get something to keep at hand for true fungus ('fungus' in the 'fungus cures' is a bacteria, called fungus because it looks like it) though I find AQ salt good for that..
A 1-gallon hospital tank
A heater is -essential- to the health of tropical fish.
Tank size.. yeah, 1 gallon is a choice some people make, but it's a pain to keep clean, lots of work, and always the risk of too much ammonia if you're sick or suddenly away and unable to clean the tank.. 2.5 gallon is what I'd call a reasonable and less hassle-y size. Fish size/shape/temperament ought to be a factor there. If you have a small floofy fish, that would probably be enough. My big plakat however would leap out and eat my face if I put him in one of those.. he's in a 10 gallon planted tank all alone. And my female is so zippy, she is quite small but occupies every inch of the 3.5 gallon, I can't imagine her being happy in 1 gallon...
High-protein food is another good call. If the first three ingredient are 'whole' fish or shrimp, or even fish meal, it's way better than useless filler wheat and corn starch etc - bettas are carnivores! People say to feed live food as treats only, but IMO bettas are healthier on live foods like mosquito larvae, fruit fly etc. Maybe blood worms and the like are too filling or whatever, but small insect live food should be a bigger part of the diet.. plus it's fun for them to 'hunt' it.
Bettas need some place to hide. A betta log is cool, but mine like plants - silk is great - NOT plastic which can rip fins - or real ones are also good (for which you'll need an appropriate light).
Good water conditioner is the last essential. I use Prime and a drop of stress coat now and then. Prime is great and cost effective as you only need a drop per gallon (I use twice the rec. dose, though, to no ill effect at all).
As for games? chase the finger (outside the tank) is a favourite. Some like to teach them to jump for food, but with my two vicious fish, it's more like 'jump to latch onto a finger, now where's my pellet dammit' and therefore not so fun..