I was the same way going the first time to WDW, but it changed me. I basically have gone on every ride there, including coasters (still need to warm up to one last ride)
Magic Kingdom:
-Splash Mountain, definitely! There is the big drop at the end and drops on the ride, but they are so quick, you'll find it easy.
-Big Thunder Mountain Railroad, It is a roller coaster but it is very tame and I'm sure you can handle it.
-Space Mountain, It is a bit more intense than BTMR and may not be for you, but I tried it out and thoroughly enjoyed it my first time. It was my first coaster.
-Haunted Mansion, it is a classic!
-Pirates of the Carribean, it is a classic!
-Peter Pan's Flight, I think I would've liked it more if I hadn't waited so long, so grab a fastpass for this
-It's A Small World, why, of course!
Epcot:
-Test Track, they advertise it as the fastest ride in Disney (which it is), but I know someone who wouldn't go on any of the coasters and loved this ride. It is just like driving in a car.
-Maelstrom, it is a mini-version of Splash, I don't like it since I love Splash, but you will probably enjoy it
-Soarin', a must-do! It is a very fun ride, no high speeds or anything, it is pure fun
-Spaceship Earth, the opinions vary on this ride, but I still like it
-Gran Fiesta Tour, there is never a wait and it is a cute boat ride.
Hollywood Studios:
-Tower of Terror, I was wicked afraid to go on this, but didn't go on until my second trip and it isn't as bad as people say, but wicked fun. I'm mad I didn't go on it my first trip
-Toy Story Mania, it is very fun, but grab a fastpass ASAP!
-The Great Movie Ride, it is a very fun experience and the ride is narrated
-The American Idol Experience!, you may or not love AI but this is a great new show
-Lights! Motors! Action!, the classic stunt show, but they show how it's done. Skip if its really hot, since it is wicked hot in this area
Animal Kingdom:
-DINOSAUR, it is not a roller coaster, you do get jostled and it does speed up, but its not scary unless you don't like loud noises
-Expedition Everest, if you were able to do Space and Thunder, do this, it is a fav of mine
-Kali River Rapids, it is tamer than Splash, but you get a ton more wet
-Kilimanjaro Safaris, probably the best safari I've been on
-Finding Nemo: The Musical, it is very cute
-Primeval Whirl, not a favorite of mine, because of the sudden jerks, but it is a very easy roller coaster, the easiest of them all
-It's Tough To Be A Bug, I love this show! My favorite show in all of Disney
I'll add the don'ts later