What kind of events would help promote my new business, and how do I find them?
Since you don't mention whether your intent is to be local, regional or national, I'll presume you will be selling to a defined local community at first.
An easy way to find local venues, trade shows, family events with vendor opportunities, as well as networking groups in your area, is to Google them. Also, visit the online version of your local daily paper and the radio stations, as they always have community news pages.
Check out any weekly publications as well. They are great for publishing press releases about your new business (and they usually love getting well-lighted photos to run with the piece).
Join a local regional Chamber of Commerce or your county's business networking group to start, and get to as many functions as possible. Call your local high school and college(s), as they may host some of the kinds of events that would be a good fit for you.
Remember that the more people who know about you, the better. Be prepared to offer discounts, and give out lots of samples; people will want to see if their dogs really like what you are selling.
Good luck with your business!