Try Omega betta food if you can find it, but keep what you have. My betta has four different types of food and all he'll touch are the Omega Betta Flakes... I'm going to have to get another fish just to use up these pellets... Oh woe is me
A filter IS a great idea, just make sure you baffle it... there are two methods in which to do that... I just use pre-filter sponge over the intake valve and a bigger sponge in front of the fountain. This will decrease the water changes a bit and make sure your betta has better water in between changes.
For equipment, my fish loves his Betta Log, so try that. Bring some nylon's with you to the store tomorrow or whenever you go to get decor. If the decor scratches the nylons it can hurt your betta. Don't worry about lookign weird, you'll probably know more about betta care than the store employees once you read a bit on this forum and you shouldn't be the first customer whose done the nylon trick.
I recently had a store employee call me a fanatic... but this was the same employee who told me bettas don't need heaters and will eat plants. Sure dude... sure. Watch your betta swim around and tell me it's not happier with a heater.
If your betta is a jumper, do not let them talk you into bamboo. The bamboo must stick OUT of the water. My betta doesn't jump- thank goodness- but you never know. What else? Since your fish is new, treat him a bit like he's sick. Change the water very frequently- just let him adjust to the PH and temperature from now on

- and put maybe a teaspoon per gallon of epson or aquarium salt into the water changes. To do this you need a water jug filled with conditioned tap water. Add the salt. Cap the bottle and shake it. Let it sit and then shake it again. Let it sit. Then change the tank water.
I say to do this because betta behavior lets us know if they are ill, but this fish is new. What is new behavior in a brand new fish? You do not want to medicate, but keeping the water hyper clean for 10 days shouldn't hurt. Plus, if you prepare for him to get sick, you'll be prepared if it happens.
Protip: Keep the cup he came in. It's a great little cup.
My betta never goes into the fish net (I just use it to remove food particles). When I change the water:
I first turn off the heater, then I use the cup to take water out of the aquarium and into a fish bowl like your 1-gallon fish bowl. Then I lower the cup into the tank and snag him that way. Then I just float that cup in the fish bowl (if he jumps out of the cup, he'll only land in a 1-3 full fish bowl. No harm ^_^ )
Then put the thermometer in the cup with the fish, I empty the tank, clean the gravel and the heater, put the heater back, dump in the pretreated water about 1-3 of the way, add the gravel back, put in the rest of the water and then plug in the heater. If the water in the fish cup has cooled down too much, I'll pour about half of the water out and float the cup in the tank with the water that is heating up. I give it about 15 minutes and add water from the tank to the cup so it's 3/4 of the way full. After another 15 minutes I pour out half of that water (into the bowl so if the fish jumps or falls he's safe), and replace the lost water with water from the tank. Again, float for 15 minutes. By this time the water is usually up to a good temperature. If not, just repeat. I remove half the water from the cup and replace it one more time, float, then slowly tip the cup. Keep some air in the cup in case the fish wants to breath, and wait for him to swim out.
That's just my method. It seems to stress out my fish very little and he's very cooperative when it comes time to change the water.
Your fish should be okay, if not a bit shell shocked from the abrupt change of not acclimating him to the ph and temperature of the bowl water, and you really only added a lot of ammonia to your bowl water by dumping in the water from the cup, but he should be okay.