Let's see what you can do with your bowl and Betta. First of all, either completely change the bowl once or a couple of times a week with water treated with a water conditioner appropriate for your tap water. A better suggestion would be to get a small gravel vacuum and do one or (better) two or three 50% partial water changes a week. The reasoning is this, although nitrogen cycles are very difficult to establish in bowl, your Betta will benefit if you can do that. Please see the source below or Google "establishing an aquarium nitrogen cycle."
Because of their size bowls are a lot harder to keep fish in than larger aquariums. Small tanks are tough to work with if the whole (fish) room isn't heated to a safe temperature for the fish. The larger the tank (up to a point) the easier it is to stabilize the nitrogen cycle and the water chemistry in general. Advanced aquarists are increasingly aware that if they don't do at least weekly partial 50% water changes (allowing for the lapse of a busy week or vacation) then eventually their fish will die of filthy water. (Very toxic ammonia after all is crystal clear.)
You probably don't have any water prepared for changes. Drop by your grocery store & buy 1 or 2 gallon jugs of "spring water." You want the water "purified by reverse osmosis" with the key minerals (calcium, magnesium, & potassium) put back into the water. (If we used water w/o minerals we can kill our fish as their gills hemorrhage from the sudden dramatic changes in water chemistry.)
If your bowl is sitting on a table or shelf set a gallon of that water next to it until the temperatures are the same. (In this case your "finger thermometer" will do the trick.)
If you have a gravel vac (Google "small aquarium gravel vacuums" or call a nearby pet shops) use it to take a lot of the dirt and feces out of the gravel. When the bowl is half empty, gently add the half back. Pour gently or pour into a small soap-less jar or use a piece of airline tubing to siphon the water in.
When you have emptied the gallon water jug, let the tap water run a few moments and fill it to the point where the surface is still the width of the jug. Treat with with the water conditioner appropriate for your water source. Leave the jug open over night in a warm place. Leaving it open will allow it to "breathe" out noxious gasses such as chlorine, free nitrogen and carbon dioxide. Tap water is usually oxygen poor and it may also absorb a little free oxygen. If you can, store the water in a warm place or at least near the bowl.
Bettas do best at 78-82 F. There are a couple of really small submersible heaters available. Among them is the Hydor MINI HEATER Aquarium Heater 7.5 Watts for Tanks 2-5 gal. Tetra also sells a small heater preset for 78 degrees F.
Those are safer, more efficient and more durable than heaters hung on tank sides. Bettas can be jumpers. It is wise to cover their bowl with a piece of scrap Plexiglas or an extra window pane or even plastic wrap. That also holds heat and potential evaporation in too.
Feed no more than the Betta can eat in 1-2 minutes It is better to feed what we might feed in one feeding over 2 or 3 feedings. That works better for humans too. ;)
There are several small books out on Bettas. Go to amazon.com, select "books" and then search for topics like "Bettas" or "Betta splendens." Highlight and copy those to notepad, save 'em and run off a copy. Drop by your public library and see what they have. Go to the reference librarian and you can have those books borrowed through inter-library loan (Your tax dollars already at work.)
Your pet shop was a little vague about what to do when changing water. But they at least got the need for frequent partial water changes right. You are right on about that too.
The person who never changes water will see their Betta sink to the bottom of the bowl and literally suffocate as the ammonia or nitrogen builds up, leaks through the gills and prevents the blood's hemoglobin from carrying oxygen to the cells.
I hope your Betta thrives and and it is a joy to keep.