UPDATE: To maybe help your design focus, we've attached an image of our mascot dogs - we love the style of that (title L&H_Mascot), so a font/logo feel that can coordinate with that would be great. I'm not sure how to word that middle ground between playful and refined, but this hit it on the nail for us.
Things that are important: a logo simple enough to be able to print in both color and black and white. Something that can translate well to a variety of settings - from signage on a building to a single color imprint on a t-shirt.
We don't care for script fonts, but we do like a font that is unique and easily identifiable with our brand, or a font that incorporates something that tells a story of who we are. See the Pizza logo attached for an idea of what we mean. We would like the business name to be able to stand out / be able to stand on it's own rather than relying mainly on an accompanying graphic included with the logo (see Birchwood Animal Hospital and Centennial Animal Hospital - without the graphic of the pets, the words are simply a generic font that wouldn't stand out on its own).
In addition to the logos for visual inspiration, I've included a couple of logos that I do like from other retailer friends (Mutts & Co and The Green Spot).
Logo styles that we absolutely want to avoid are over the top, cartoon-style logos (woof n whiskers), and a logo with a dog and cat intertwined (pet logo).
Please ask questions if you have them!