A happy, healthy, dog should be the focal point. I've attached an image of my dog. Don't make it a caricature, or even consider including her to be mandatory, but it should serve as some inspiration.
The overall logo may be larger, but it should focus on a central image that can easily serve as an icon, profile image, sticker, etc.
The graphic as a whole should easily translate to black and white, or at least one-color printing. Imagine something that could be made into stamp.
Google, "Chalkboard Logos", or "Vintage Whiskey Labels".
At the moment I'm a big fan of "DOGFISH" in Nexa Rust Sans Black font. Not imperative, but worth looking into.
In general, I'd suggest avoiding any imagery that includes fishing hooks. Feel free to try, but my gut feeling is that they don't convey a very friendly brand image.
Although DOGFISH begins as supplier of fish treats for dogs, we have every intention of branching into other types of food. So don't get tooooo fishy :-p
I'm also attaching some of my own ideas as they come up. Don't let me make you guys look bad!