If you head left at the start of Level 1-1: Mangrove Cove, you can enter the airplane and find static-filled TV screens. If you Ground Pound beneath them, they will display the Donkey Kong Country Returns title screen and play its opening theme song.
When you shop at Funky Kong's Fly 'n' Buy store, his dialogue for the Crash Guard item includes, "It's dangerous to go alone take one of these!" This references the famous line, "It's dangerous to go alone! Take this" from the original Legend of Zelda game.
Opening Theme Song
If you continue past the ending barrel in Level 1-3: Canopy Chaos, you can destroy a box to find a jump spring. It will take you to a secret platform. On the middle platform, use either Cranky Kong or Dixie Kong's abilities to make it to another platform above you and to the right. You will see the record player Cranky had at the start of Donkey Kong Country. Ground Pound beside it and it will play the Donkey Kong Country opening theme song.
The platforms in this section are from the original Donkey Kong.
Donkey Kong's idle animation when you haven't done anything for a while is to get out a 3DS and start playing it. He will either play Animal Crossing: New Leaf, Mario Kart 7, Super Mario 3D Land, or Donkey Kong Country Returns 3D, which you can identify by their music. He will play multiple 5-second sections of each game.
In Secret Level 1-B: Busted Bayou, Samus's Gunship from Metroid is in the background around where you find the letter K.