What is the best way to treat orthodontia-caused blisters on my gums?

Ask your ortho. Determining the cause of "blisters" on your gums requires personal examination of them by your orthodontic specialist. Orthodontics is not typically a cause of blisters. There is the probability you are describing either oral herpes (cold sores) or canker sore (aphthous ulcers). Warm salt water rinses can provide some relief until you see your orthodontist.
Depends. In large measure, orthodontia is not the cause of oral blistering. It is common for cheek and lip irritation to cause small sores at the beginning of treatment, until the skin or oral epithelia is calloused to the irritation. Blisters, lesions containing liquid or serum are usually a manifestation of a herpetic viral outbreak, and orthodontics at the start can be the trauma that initiates it.
Prevention. There is a specially formulated wax for use orally in orthodontics. It ise is to place on sharp brackets and wires which blister oral tissues. Most orthodontics recommend its use for prevention of oral blisters caused by the hardware in your mouth.
Depends. Are the blisters due to the braces rubbing against the gums? If so you can place wax until the gums heal. Sometimes you can get blisters due to the reaction of the dental materials. If so the type of materials/braces can be switched out.
See dentist ASAP. Braces do not cause gum blisters unless you have a true nickel allergy (rare). Gum "blisters" may be from abscessed teeth, gum disease, impacted food debris, or a myriad of other causes. See your orthodontist and your general dentist to diagnose the cause and then resolve the issue.
Ortho wax. There is a special wax formulated for use orally in conjunction with wire/ bracket orthodontics. It's use is to place it on the offending sharp wire or bracket to protect the delicate soft tissue of the patient. It is very effective and most orthodontist will direct their patients to its use.
Protection. If there are any sharp areas on your braces or brackets, see the dentist for adjustments. You can use special wax that your dentist can provide to cover any areas of the "hardware" that are causing discomfort.
See the orthodontist. Perhaps some of the hardware or appliances or trays need adjustment as they may be irritating or rubbing tender areas in your mouth. Your orthodontist can supply you with remedies.