Let me clear up some things since these answers have left out information.
The quickest way is the bus -- you get it at the Main Entrance (straight ahead and on the main entrance route) Park to Park takes about 10 minutes -- but you can have up to a 30 minute wait in between busses at certain times of day. Less around lunch and dinner times. This is by far the closest dropoff spot to Coral Reef.
The next quickest way is the boat -- you pick it up directly to the left of the main entrance as you exit -- it takes 30 minutes (not 20) and it makes stops at three resorts but continues on to the International Gateway (i.e. back entrance by France) at Epcot. That means a long 15 minute walk to the front of the park and your Coral Reef dinner.
While it is possible to walk from DHS to Epcot, it's a very very long, and very hot walk -- and then another 15 minutes to the front of the park. Don't do it.
I've done this trip many many times, and I would highly recommend you just take the bus. If you had dinner reservations in World Showcase, I would recommend you take the boat. I would never recommend you walk.