That will probably be the fins shredding, from those horrendous spiky plastic plants (fins are really delicate..and bettas love to shove themselves through the plants like little underwater badgers sometimes).
3 gallons is a great size. Dunno if they'll come with a heater, but a good, adjustable mini-heater is a great investment. If you can afford it, get a spare as well! (I just -melted- my spare, onoes! )
Sponge filters are safe, those are what I use, but maybe somebody can recommend a safe, quiet alternative with perhaps an internal pump (external ones can get a bit annoying if they are near your bed).
Silk plants. Those are nice, and won't shred your fishy's fins.
A bottle of Prime dechlorinator.
A gravel vacuum, for water changes, and a plastic bucket or two that's only for use with fish water.
Keep the small bowl as a hospital tank in case of sickness.
A bag of aquarium salt, a box of unscented epsom salts, and a bottle of broad spectrum antibiotic, for just-in-case. Better to have it handy, than not have it when you need it.
Those are my basic 'essentials'.