Many Scouts ask, "what are the easiest/fastest merit badges to earn? From what we've seen, it really depends on your interests as a Scout. Some Scouts find some merit badges easy to do, and some are challenged by them. The fact that these are badges you EARN, means that these badges all have some challenge to them.
Bryan on Scouting ( a favorite reference) talks about first badges here: http://blog.scoutingmagazine.org/2016/01/26/what-makes-the-best-first-merit-badge-for-a-boy-scout-to-earn/
Just a reminder, don't let more challenging (Silver/Eagle) merit badges scare you - doing a little bit, a small requirement, towards them every week adds up quickly!
Here are some merit badges to help you get started quickly:
Don't forget to log your information into SCOUTBOOK as well as getting your blue card signed!
Animation (5 requirements) Takes about two weeks.
Art (5 requirements) Takes about a week.
Basketry (3 requirements) Takes 1-3 days. A summer camp fave!
Communication (9 requirements) Takes about a week to a month
Family Life (7 requirements) If you start during the winter holidays, you'll earn it by Spring!
Fingerprinting (5 requirements) Takes about a week.
Golf (8 requirements) Takes anywhere between 1-4 weeks.
Indian Lore (4+ requirements) Takes about a week
Landscaping (6 requirements) Takes about a week
Leatherwork (4 requirements) Takes about a week
Moviemaking (4 requirements) Takes 1-4 weeks
Music (4requiremnts.) Takes 1-6 months, not weeks. Start now, and be done by summer
Reading (4 requirements) Takes 1 -4 weeks
Scouting Heritage (8 requirements) Takes a week to 8 weeks.
Sculpture (4 requirements) Takes one week
Textiles (6 requirements) Takes a week
HOW TO USE A BLUE CARD: https://monmouthbsa.org/wp-content/uploads/2014/07/Merit-Badge-BlueCard.pdf
Need a blue card in a hurry? See - http://usscouts.org/advance/docs/34124.pdf
Good Luck!