First of all congratulations!
My husband and I have taken 3 anniversary trips to Disney World. We love the mixture between being able to relax at a beautiful resort and being able to enjoy the romantic/ fun dining and the high energy of the parks. In fact, we love it so much that last year we bought DVC points so we can take an anniversary trip every year more affordable (since a romantic Disney trip is about much more than the parks).
My first suggestion is to plan well and make reservations ahead of time without planning every spare minute. I learned after the first year how important it is to make good reservations ahead of time if you want to eat at the romantic/ higher class restaurants (which you can book 180 days ahead of time) and how important it is to NOT cram in as much park time as humanly possible.
When I start planning (180 days ahead of time...okay 179...I'm somewhat OCD), I write down each day we will be there, and next to it, I look at the calendar for each park and write down when each park is having extra magic hours. Then, I decide what park we are likely to be in around dinner time. My husband and I really only use the night extra magic hours at Magic Kingdom (and occasionally Epcot) and the morning hours at Animal Kingdom (and occasionally Hollywood) because hours at the other parks at the other times are WAY too crowded. Also, we never spend the whole day in a park that has extra magic hours that day because it is WAY too crowded. The parks are usually the least crowded on the day after they have had extra magic hours (particularly evening extra magic hours), so we usually plan to go to the least crowded park (though we sometimes do dinner at Magic Kingdom when they have evening hours.
After choosing the park we will be in around dinner time (or the place...don't overlook a nice slow day at the resort or a fun evening at Downtown Disney), we decide which restaurant sounds best there. We have some that we LOVE and go to every year, but we enjoy branching out and trying new places too. Remembering that it can take up to 1 1/2 hours to get to a dining reservation from the resort or from somewhere else, I find the time that is most convenient for us to eat. After determining which restaurant we will go to each day, I call and book the reservations.
Some of our ABSOLUTE favorites are Coral Reef (seafood and steak in a romantic atmosphere with a huge aquarium that makes up an entire wall of the restaurant...we always wait to sit right by the aquarium, and we make sure they know it's our anniversary), Le Cellier (VERY romantic, and the fillet mignon with risotto is excellent...it cost one credit for lunch and two for dinner, so we always eat there for lunch), Raglan Road (a fun, beautifully decorated Irish pub with live Irish music and Irish dancers that starts at 7:00...we usually spend the evening here and take advantage of the fact that we don't have to drive home

...more fun than romantic, but it's a blast!), and Mama Melrose's with Reserved Seating for Fantasmic (quaint Italian Californian fusion that will save you from having to wait for 2 hours to see the BEST show at any of the parks...I have also heard the Hollywood Brown Derby is good, and they offer this too but for two dining credits). All of these options are romantic or fun and cost only one dining credit. I would not feel like our anniversary trip would be complete without any of these.
This year, we will try Tony's (since we have decided to pay for one meal out of pocket to use two dining credits on a Signature restaurant), Rainforest Cafe in Downtown Disney, and The Dining Room at Wolfgang Puck's (2 credits). In the past, we have tried The Wave (which we both agreed was just alright), Chefs de France (which has delicious food but which we found to be WAY too crowded), The Crystal Palace (which was fun, but we're not buffet people), Cindarella's Castle (which includes pictures and gratuity for 2 credits, but which--in my opinion--was equal to the princess dinner at Akershus in Epcot that also includes pictures and only costs one credit), and The California Grill (which to us seemed over-priced at two credits since we preferred the food and atmosphere of Coral Reef and LeCellier to it), and several other restaurants. In the future, we would like to try Bistro de Paris and wait for a table near the window with a view of Epcot's fireworks (this costs 2 credits, but we have heard it's worth it).
Aside from planning and restaurants, I strongly recommend taking time to relax at the resort, lounge at the pool, putz around Downtown Disney, and just...in general...chill. I do all of my planning (which realistically does not take that long) ahead of time so that we can do whatever we want and be spontaneous while there. "No planning at Disney" is a pretty good motto to live by (at least while on vacation

). Knowing which park is least crowded has made our trips MUCH more relaxing too.
One more fun place to go is Jelly Rolls (a dueling piano bar where they can play ANYTHING you request...have some singles to accompany your requests, and again, we take advantage of the fact that we don't have to drive). There is a $10 per person cover charge, but it is FUN! We always walk there when we leave Hollywood, and the walk is romantic since it's nice to get away from the crowds...we have only ever encountered 2 other couples on the walking path there.
As far as romantic resorts, we have enjoyed all of the ones we have been to. The quiet atmosphere, fun pool, and easy walk to the parks make Beach Club a fun and romantic experience. The Savannah View rooms at Animal Kingdom made it fun to sit on the balcony and relax with a glass of wine or a cup of coffee. The boat ride to Downtown Disney and the pedicure my husband surprised me with at the spa made Saratoga Springs convenient and relaxing. We liked the boat ride so much that we decided to try Old Key West this year (since it offers a boat ride too).
We have always paid for a studio room (which is fine), but twice we have been upgraded to a one bedroom since we asked upon check-in and told them it would make our anniversary special if they had any upgrades available. The one bedroom suites have a living area and a whirlpool tub! (I always go to Basin...a store in Downtown Disney...and buy one bath bomb for each day. This would be a nice surprise for your fiance since they deliver these to a desk at your hotel, and you wouldn't even have to carry them back with you.) I would recommend booking a one bedroom if you can afford it and if you will be spending a lot of time at the resort. They will likely upgrade you from a studio to a one bedroom, but you can't count on that. That being said, the difference in cost is pretty big, so I always tell myself (lie...lie...lie) that I will not be disappointed if they cannot upgrade us. It certainly never hurts to ask though (and make your wedding known).
Well, that's my book...I am not usually so long-winded (though I do teach English, so I will blame my college for my gratuitous phrasing). Have fun, and relax! I hope your honeymoon is romantic!
