Estuarine Bio degree and lifetime freshwater/brackish/salt/reef/whatever aquarium owner here. Two main things will make fish ill and/or kill them. Ammonia and shock.
Ammonia is caused by organic things rotting in the tank - food, crap, etc. Feed your fish every other day, or even third day, and make sure they eat everything. Make it a goal to have no food go uneaten. Otherwise it will just sit on the bottom, rot, and turn into ammonia - like the crap. There are beneficial bacteria in the bowl/tank that will convert that ammonia to nitrite/nitrate, but without a filter, there is very little surface area and aeration to promote bacteria and it's respiration. The end result is since there isn't much bacteria, it can only convert so much of that poisonous ammonia before it starts to build and exceed safe levels. Fish gets stressed from this, gets sick, dies. Cut down on food and you can go a week or two easy without a water change.
The other common shock is the water change. It's very necessary, but needs to be done right. First off, let the water come to room temp that the goldfish's water is already at. Second, get the chlorine out of the water (if it's from tap). Either let it sit for a day to let the chlorine naturally dissipate or buy drops that will remove it immediately. No need to change all of it - try half.
That's it. Feed less, change water as needed (weekly is good for a fishbowl without filtration) and make sure it is the same temp as existing water and chlorine free.
