For breakfast it's the Wicked Oyster and for lunch it's Mac's on the wharf. The Wicked Oyster is bright and loud enough that a few well-behaved kids aren't going draw much attention. They offer a wide selection of breakfast items that should satisfy anyone. I especially recommend getting the berry topping for your french toast, or going with the omelette special. The portions are generous, but not ridiculously so. They also serve lunch and dinner, but I haven't yet had the chance to sample those.
Mac's on the other hand is classic beach food joint. Order and pay at one window, pick up around the corner and either eat at the picnic tables or take it back to wherever you're staying. I have to admit that I probably shouldn't look forward to Mac's as much as I do. To be perfectly honest, the "Mexican" rice is at best bland, it's always busy so you're going to be waiting at least 10 minutes for your food and it's a chore to find a free picnic table that isn't downwind of the garbage bins. On the other hand, the fresh seafood (did I mention Mac's is one of the local fishmongers and you can buy fresh fish next door if you want to prepare it yourself?) more than makes up for it. They also make pretty decent onion rings (second only to King's Roast Beef in Salem, MA). The scallop burritos are excellent, the aforementioned "Mexican" rice notwithstanding, and the Portuguese kale and potato soup is a must try. I think it's more positive association than the food. There is also Mac's, just down the road toward town, and that is a traditional restaurant, and, as you'd expect, the food is slightly more sophisticated, but as with most of the town eateries, it's still a pretty casual place.
You can get ice cream at Mac's on the wharf, but unless you need to have it right now, or for some reason you need to eat it by the water, I'd suggest the ice cream shop next door to the downtown parking lot (seasonal). Try the Wellfleet blend.
Next door you'll find Winslow's Tavern. They do a nice lobster roll (among much else), and I'm always impressed when a place takes the time to toast or grill the roll. That simple step takes a lowly hot dog roll and elevates it to something special (May-October).
We've never eaten at The Bookstore, which is right across the street from Mayo beach. Any time we've tried, there is always a wait of at least 45 minutes. It's certainly popular, and, by all accounts, serves pretty decent food.
The Juice is a small place that primarily serves wraps just on the way out of downtown toward Mayo Beach and the wharf. It's in an old house that has acquired a bit of a slant over the years (see the photo) and is a nice place to take a break from the art galleries, with indoor and outdoor seating that is conveniently located to many galleries.
If you're looking for more of a romantic setting for dinner, Terra Luna is just out of town (technically, I think it's in Truro) and serves excellent italian-style food. We were able to ditch the kid with the parents last summer, and enjoy a nice quiet dinner. The desserts are also excellent.
Another very popular place is The Wellfleet Beachcomber at Cahoon Hollow beach. Unfortunately during the day you'll need to pay to park, assuming there is even a free spot to pay for, since there isn't any free parking within walking distance. It's open late, so if you're looking for some nightlife, this is one option.
Wellfleet is chock-full of good places to eat, without a major chain in sight. If you are craving a Starbucks, you can either drive back to Bourne (about 45 minutes or much more), or you can get a good coffee, espresso or any other coffee-shop beverage at The Juice (seasonal), The Lighthouse, the Flying Fish Cafe, or the Wicked Oyster (use the back door for direct access to the coffee bar, avoiding the restaurant crowd out the front).