Is it necessary to become bonded if I want to start a dog-walking business on a small scale? If so, how do I become bonded and under what category?

What you may be referring to is Liability Insurance, not Bonding. Although it may be a good idea, if you are going to do this on a "small scale" you may not have to consider it immediately, possibly after your company is larger. Liability Insurance is not cheap.

Are you incorporated? If you plan on growing your company, that would be my first priority.
Stringer