Hi, never been to Skillets, so can not comment on that one. Always seemed kind of touristy ? to me. For breakfast, we like either Signes, Sunrise Cafe, Grubys Deli, or Harolds.
Harolds is very very casual, has great omelets, and "grill type things" Sunrise Cafe has a more varied breakfast menu, and is usually packed in the AM. The patio has the big outdoor patio heaters, so still great to sit out there, most days, even this time of year.
The other place for breakfast we hit every once in a while is Stacks, but not one of my favorites –but my husband loves the buffet.
hope that helps on the breakfast places.
Another place to mention, not for breakfast, is the Sea Shack. We usually do a carry out there, since the atmosphere is pretty much non-existant. But, last week I had the Tilapia from the grill and it was some of the best I have had on the island. Don’t let the looks of the place fool you, the food is good.