Burnout skills are one of the easiest skills to do in the game. Simply get into any car with a good amount of power. The more power the easier it will be to break the tires free and initiate a Burnout skill.
I found the easiest method for me doing these Burnout challenges is to use an automatic. Simply floor the gas to get the first Burnout and then after 1-2 seconds slam on the brakes. Hold the brakes until reverse kicks in and keep on it to get a backwards Burnout. Then slam on the gas for a forwards Burnout and repeat until done.
Just make sure you are on pavement to get these to work as I don't think you can actually get burnout skills off road. At least I never have.
Also keep in mind that there appears to be a slight delay in Burnout skill acquisition as sometimes my skill wouldn't count. So you will likely have to do more than the required 15.
Manual, and with clutch, will work I just never use them if I don't have to so I don't know how fluidly and quickly you can go from forwards to reverse to forwards.