Tap water is definitely best because it has minerals in it your fish needs and distilled does not have this. Don't use distilled! Peyton go to Petsmart and buy a water conditioner called Prime this should take care of the chlorine. Since you have more chlorine than normal I would suggest making your water a few degrees warmer than needed and let it sit for a few hours until it reaches the temperature of your aquarium this will give the Prime a chance to do the job more thoroughly. I have purchased a Top fin 7.5 watt heater from Petsmart and I have been very happy with it for heating a 2 gallon container. It should work fine for a 2.5. Test the heater for 24 hours in a similar size container to be sure it's working properly before using in your aquarium. Make sure you get a thermometer. When you do your next water change make the water about 80-82 degrees and put the heater in the aquarium. Have your fish cupped and let him float to acclimate to the new temperature.
As for cycling you could do it with your 2.5 gallon. Some people don't worry about it for small tanks. Cycling is best done without the fish in the water due to amonia spikes. However it can be done with fish in but your have to monitor your water constantly and do changes more often. I have done it rather naively not knowing better my fish are fine but there was a point where one the ammonia spike and one of my fish was definitely sick but I changed the water quickly and she is fine now.
I would suggest if you can afford another tank Petsmart sells 2.5 gallons for $13 doing a fishless cycle. You can use an extra tank for a hospital tank or another fish later.
You will need a water test kit which is cheaper online:
You should have a test kit whether you cycle or not to help keep your water quality good and your fish healthy.
I have this sponge filter in my cycled tank:
The sponge moves up or down so you can provide more room for you fish depending on where your fish tends to swim. You will need an air pump Petsmart sells one by Topfin for 10 gallon or less than ten gallons. If you want more fish get the 10 gallon if not get the one for less. You also will need check valves which saves your air pump if the power goes out, airline and T valves.
If you don't cycle you might look at this filter:
While you are ordering Amazon also sells a food that is high reccomended for Bettas:
If money is an issue take you time getting what you need. Just get your heater and keep your water change schedule. Then over the next couple of months buy what you need. My fish went without a filter for a short time and they were fine.