Also, being a smaller town some restaurants will close for a day or two mid-week and may only be open for dinner, especially in the off-season. Most of the higher-end restaurants have a wine bar and many have outdoor seating. These restaurants are on the town square or within a couple blocks. There are more restaurants in the downtown area but these are the ones I know.
if your stomach is man enough
On the cheap
There are several Mexican taquerias in town. Near downtown are El Farolito and El Sombrero.
Flying Goat Coffee. Excellent breakfast breads and great coffee.
Bear Republic Brewery. An extensive list of salads and burgers plus their own outstanding beer. Jamaican Jerk Burger and a Red Rocket Ale, mmm. Indoor seating is noisy; outdoor seating is nice. A good place for the kids while dad relaxes with a cold one.
Costeaux French Bakery for a light breakfast or lunch. Costeaux has some of the best breads that can be found anywhere.
Downtown Bakery & Creamery is another good breakfast or lunch stop.
Healdsburg Bar and Grill. Salads, burgers and beer and a great outdoor seating area. Nice list of local wines to go with your burger.
Oakville Grocery. A wine country chic grocery and deli with outdoor seating.
Barndiva. Locally-sourced foods and an extensive cocktail lounge (you don't often find good cocktail lists in wine country).
Charcuterie. Sometimes called "cute" or "romantic," but it definitely is small.
Ravenous. I call it wine country home-cooked comfort food. They've been around a long time and a favorite with the locals.
Scopa. Small and family run. If my mother was Italian I could only hope she could cook like this.
Willie's Seafood. Good spot for lunch of a light dinner as everything is served in small plates. The outdoor patio is very popular at lunch time.
Zin. They call themselves "American food with fresh local ingredients." Large dining area and a favorite with visitors.
A bit higher
Dry Creek Kitchen. A Charlie Palmer restaurant. If you feel like dressing up a bit for dinner (though you don't have to) this is a good place to go. The food has a sort of a "back East/NYC" flavor to it rather than the typical California/Mediterranean-style of many area restaurants.
Sky's the limit
Cyrus. Two stars from Michelin. Cyrus and The French Laundry are considered the best wine country restaurants. I've never been to either as I don't have the required six-figure income. People who have been there call it not just dinner but an experience. You can also check out their excellent bar and bar menu where you can have a couple drinks and some bar food and escape for about sixty bucks.
Where to take the kids
The more kid-friendly restaurants would be the Bear Republic Brewery, Center Street Cafe, Downtown Bakery, and the Healdsburg Bar & Grill. Outside of the downtown area are Giorgio's and Singletree.
For what it's worth, if I'm in town for a day you'll find me at the Flying Goat for breakfast, The Bear Republic for lunch, and Scopa for dinner. Then I'll spend the next two days at the gym!