Un treated abscess from tooth decay do not spread to adjacent tooth but infection potentially can spread to other area, this condition called cellulitis and that is a medical emergency.
Abscess from gum disease do not spread once you taken the right antibiotics.
To facilitate healing you should keep the area clean with tooth brushing, flossing, use mouthwash and finish the whole course of the antibiotics. It is also important to come back for the Dentist follow up, and provide any others necessary treatments.
Healing will require a root canal treatment and subsequent crown of you want to hold on to that tooth long term. Your other option is extraction and bridge of healthy teeth or an implant into extraction site.
Once on antibiotics they're isnt a whole lot you can do to make it better but I always recommend to take multustrain probiotics at least 2 hrs after you take antibiotics this week help minimize detriment to your gut flora