Tail rot or fin rot
•Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter or they seem to be falling apart and dissolving, Black or red along the edge of the betta’s fins/tail, Bloody tips, Behavior may not change
Treatment: Conservative: Treat with Aq.Salt at 1 tsp/gal. Increase water changes to 100% daily. Replace accurate amount of salt following water changes. Add Stress Coat to help repair tissue. If there is little to no improvement within the first 5 days, you can increase the salt dosage gradually to 2tsp/gal but do not continue any salt treatments past 10 days. Medication: If Conservative treatment is ineffective use API Tetracycline, API Fungus Cure, API Triple Sulfa, OR API Erythromycin. Also add Stress Coat to help regrowth. Continue until fins/tail stop receding and start showing some new growth.
This is copied directly from the sticky about betta diseases and treatments. I don't know if you have any aquarium salt, but if you do, I'd suggest putting some of that in there at the dose recommended above. If not, I think daily 100% water changes would certainly be a good thing. That seems to be one of the very best things for a great many issues with bettas... clean water really helps them thrive and get over anything that is ailing them.
I'm sorry I can't help more, I really am rather new at betta ownership as well. But going off of what people have said on here, the treatment in the sticky seems to be the best treatment. You could always try PMing OldFishLady, she is amazingly helpful and extremely knowledgable. Also, if you can fill out the form (I'm copying it for you here) that is stickied in this section of the forum, more members might be able to offer a bit more insight for you.
What size is your tank?
What temperature is your tank?
Does your tank have a filter?
Does your tank have an air stone or other type of aeration?
Is your tank heated?
What tank mates does your betta fish live with?
What type of food do you feed your betta fish?
How often do you feed your betta fish?
How often do you perform a water change?
What percentage of the water do you change when you perform a water change?
What type of additives do you add to the water when you perform a water change?
Have you tested your water? If so, what are the following parameters?
Symptoms and Treatment
How has your betta fish's appearance changed?
How has your betta fish's behavior changed?
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Does your fish have any history of being ill?
How old is your fish (approximately)?