Well let’s see. You need to make a distinction between neat/clever and effective regarding choosing a name in that they are not always the same thing. What do other successful people do? You can go a number of directions per the list I culled from Google below.
You can apparently go “clever-colorful” or “authorized-certified” in your naming approach or combine them.
“Certified-Authorized” approach will need your name to end in Institute, Academy, Center, School etc.
“Clever-Colorful” will involve some kind of acronym like “Pro-tec” or a colorful name like “Thunder Ranch”
Combo examples are “Top Gun Training Centre”, “Gunsite Academy” etc.
If you are going to serve the general public (as you seem to indicate) a Certified-Authorized approach is probably best with perhaps a regional name ie your county, state, region, or some more colorful geographic identifier attached. If firearm enthusiasts are your market a Clever-Colorful name might have more appeal like “Lethal Force Institute” "Tactical blah blah blah etc.
According to your profile you are Tampa, Florida based so maybe
Sun Belt/Florida/Tampa/Tampa Bay/Atlantic/Tarpon Springs…etc.
Firearms Training
Gun Training
Firearms & Training
Self Defense Training
Institute/Academy/Center/School
as an example
more regional
Sun Belt Firearms Training Academy
More local
Tarpon Springs Self Defense Training School
Intermediate
Tampa Bay Self Defense Training Center
If you plan to sell the school at some point keeping your personal name off the school name would probably be a good idea.
Cumberland Tactics Firearms Training
Gunsite Academy, Firearms training in Paulden Arizona
Top Gun Training Centre
Advanced Tactical Group, Ltd
Eastex [TX]
Firearms Academy of Seattle [WA]
Front Sight Firearms Training Institute [CA]
Gabe Suarez Tactical Training [CA]
GMJ Enterprises [NV, UT]
Lethal Force Institute (Mas Ayoob) [NH]
TechCom Training
Inland Empire Firearms Training
American Firearms Training And Tactics
Tactical Firearms Training Institute
Brenzovich Firearms & Training Center
MidSouth Institute of Self Defense Shooting
Thunder Ranch Firearms Training
Pro-TEC Firearms Academy
Northwest Firearms Training Center
DEFENSIVE COMBAT ACADEMY