If you have more than three, or some unusual allergies, special diets has a form you can fill out if you email them.
Rarely (for us that means 2 out of maybe 100 reservations with an email) a chef will call you. Honestly one of those 2 times the chef then volunteered that he would make DS brownies, we told him, and when we got there he had no idea and they offered him the Enjoy life ones he is allergic to- so I'd rather they don't call now.
Mark it on your reservation, let them know when you arrive. When you get to a quick service location ask for a chef or manager if the allergy menu they give you does not have enough information.
Safest sort of depends on the allergy. We are more careful in EPCOT, language and the fact that most locations are not actually Disney owned means you have to be smart here. If you are allergic to corn, beef and dairy- the mexican place may be a bad choice. Same thing for gluten/dairy- maybe the place in Italy isn't for you.
For the most part, table service can help you out. The more diversity on the menu- the more raw ingredients they have to work with. Buffets can actually be good for that reason.
IF you happen to have issues with all of gluten/egg/dairy/soy - breakfast can be an issue. Resort quick service (with exception of YC/BC/Boardwalk) will be able to help you, but those allergies are not good for getting any quick service breakfast in parks. Lunch and dinner you should basically be able to go wherever, but you want to go for menus that have more items close to what you can have. I used to say stick with Cosmic Rays, Columbia Harbor house (unless it's shellfish), Sunshine Seasons and Flame Tree BBQ- but they have been taking measures in the past year to spread out the allergy options so it's more fluid now. It all really depends on what you can't have. We are top 8 +sesame + buckwheat + cottonseed and life is just much easier at table service. We are allergic to all the pre-packaged allergy items so everything has to be fresh.