There are lots of dining options at WDW -- and, quite frankly, shorts and sandals are fine at all of them except Victoria & Albert's. I know the "signature restaurants" have a so-called "dress code", but no one follows it. Things have become so relaxed that men now wear cargo shorts and sandals to fancy Food & Wine Festival dinners at places like California Grill -- whereas a few years ago women wore fancy cocktail dresses and men wore suits. Basically, we have seen the end of dressing up for dinner, IMO.
Putting all that aside, ask yourself do you want a more expensive and fancier dining experience or do you want something fun and diverting.
If you want fewer kids, some romance and tasty food, skip the Disney signature restaurants and go to Shula's (steak), Blue Zoo (seafood), Il Mulino (Italian) (there will be more kids at Il Mulino than the other two). Make a reservation for 8:30 pm or later.
If you want a nice restaurant but want it to be Disney, go to a signature restaurant but be prepared for lots of noise and mayhem with noisy kids everywhere and little chance for romance. Also, desserts are awful at all of the signature restaurants now (thanks to the
Disney Dining Plan (
DDP) where everybody is entitled to a dessert with dinner, so desserts are poor at best). You'll do better sticking with appetizers and entrees. Appetizers are not included in most DDP dinners so they tend to be more creative and nicer than the desserts. The signature restaurants are California Grill (noisy with tons of kids due to good view of the fireworks, pretty good food but not as good as it used to be, IMO), Flying Fish (good food and generally not as noisy as CG), Narcoosee's (fairly tasty food, fireworks, lots of noise), Citricos (very good food, lots of noise), Jiko (pretty good food but not as good as it used to be (IMO), less noise - usually but not always), Yachtsman Steakhouse (pretty good food, very few menu choices, lots of kids), Artist Point (pretty good food in a very pretty setting (I've heard rumors the food has improved of late), plenty of kids but fewer than at CG), Le Cellier (some love it but we wouldn't bother given its the same price as some nicer choices and it's very noisy). Others can fill in if there are any signature restaurants I forgot.
If you want a unique Disney experience and don't care so much about food quality, try any of these depending on the type of experience you want: Coral Reef (dine while watching large fish in a huge aquarium), Be Our Guest (dine with various special effects from Beauty and the Beast), Marrakesh (dine on middle eastern food and see a belly dancer), Cinderella's Royal Table (dine on poor food in the Castle while meeting the princesses), Prime Time Cafe (dine on mom's home cooking with nostalgia and old TV shows), Sci-Fi Dine Inn (dine in a "car" parked at a drive-in while watching sci-fi trailers), Hoop De Doo Review (dine on pretty poor food in a western atmosphere while watching a cute musical revue), Luau (dine on okay food outdoors while watching a Hawaiian revue), Sanaa (dine on tasty Indian food while watching animals on the savannah), San Angel Inn (dine on interesting Mexican food while watching boats go by or in the sister restaurant while watching Illuminations fireworks show), Rose & Crown Pub (dine on English cooking -- if dining at dinner, try for patio seating for Illuminations).
As you can see, there are lots of dining choices, so you just need to decide what you desire when you indicate good, romantic or fun.
Enjoy!
and we had our 10th anniversary dinner at Artist Point, and it was great! We did get a little dressed up, but you don't have to wear a jacket or anything. Just a nice shirt and pants would do. 
When they brought out my dessert, they had written Happy 10th Anniversary in chocolate across the plate. Basically, there were lots of little touches that helped make dinner more special.
