Hi and Congrats!
My husband and I went on a 10 year anniversary trip in September 2013 just the two of us so I guess it is was like a second honeymoon. It was wonderful. It was so fun to do Disney 'adult style' without the kids. There are a so many romantic things to do at Disney and no matter what you will have fun! Here is my unfinished trip repost if you want to sneak a peek
http://disboards.com/showthread.php?t=3177687
Can I ask what hotel you are staying at (on property or off)? Do you have free dining or are you paying a la carte?
We actually stayed at the Polynesian so had access to the monorail which was fabulous for heading back to the resort for an afternoon nap

. Highly recommended by the way, very romantic thing to do!
We started by choosing which parks we wanted to visit on which day using Touring Plans. Then I reserved all the dining reservations (mostly for dinner) and fast passes ahead of time, but once there, we just went with the flow. Some mornings we slept in or we headed to the hotel restaurant for breakfast with no reservation and asked if they had room for two. They always did. One day we even skipped the park all together and just enjoyed the pool. Then we headed to MK for the fireworks. So my suggestions is make a tentative plan but then once there just do what you feel like doing. Take it slow and don't try to fit everything in on this trip. You are there to enjoy each other's company, not to be running around like chickens with no heads.
Epcot is by far the best park for romance with the country pavilions like Mexico, Italy, Japan, France, UK. We enjoyed walking around the pavilions especially at night and discovering hidden treasures in their shops. We also enjoyed a few drinks along the way, my favourite being the Grey Goose slushy in France

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We ate at some of Disney's signature restaurants like California Grill. Great views of MK and really good food! The best evening was our dinner at Victoria and Albert's. It was expensive but worth every penny. The food and service were stellar and it felt special to dress up for the occasion. We also really enjoyed some of the less fancy restaurants too. My favourite being La Hacienda Di San Angel in the Mexican pavilion. Very romantic at night and great margaritas! (Le Cellier and Chef de France also is very romantic in the evening). We also liked our lunch at Sci-Fi Dine Inn and dinner 50's Prime Time for a more playful meal. Unfortunately I had a bad experience at Brown Derby. The atmosphere was great but the food, not so much! And we also enjoyed Ohana' with views of the fireworks that night. We did one character meal at Tusker House for brunch and really enjoyed it but did not want to have more interaction with characters than that. I loved Be Our Guest in MK so try to get reservations for that or just wait in line for their QS. It is well worth it.
I would suggest to skip the afternoon parades which are geared more towards children but not to miss any of the evening shows like Wishes, Illuminations or Fantasmic which are all so breathtaking. Take advantage of extended evening magic hours if staying onsite and go do the most popular rides after hours when the crowds disperse.
Downtown Disney and the Boardwalk has tons of shopping and entertainment at night so go check out those if you have time. You can go grab a drink at a pub or go dancing.
Even if you are not staying onsite, try to explore some of Disney's onsite resorts. The restaurants and the grounds of the deluxe resorts are spectacular. For example go to Animal Kingdom Lodge for dinner one night and enjoy the stunning lobby, african artwork and the animals right at its doorstep. Or head over to any of the Monorail resorts from MK or Epcot. Polynesian, Grand Floridian and Contemporary also have a boat launch from MK which is really nice at night.
Hope that helps, have fun!