Basically the Rose Rabbit can be found on three different locations, always after 6 pm.

Day 3 - 12 - 21: East Village
Day 6 - 15 - 24: Breezy Meadow
Day 9 - 18 - 27: Oceanfront

The food will be requested in this order.

1. Savory Dumplings (can be bought from Aroma Bakery in Spring)
   
2. Creamy Potato Soup
   
3. Corn Souffle
   
4. Steamed Lobster (You'll need a landing net to get a Coral Lobster, or you can buy these at Wake's market stall during the Autumn Market. This is also a reward by a Sage Glop.)
   
5. Smoky Deviled Eggs
   
6. Pasta Primavera
   
7. Supreme Pizza
   
8. Sweet 'N' Sour Shrimp (You'll need a fully upgraded landing net to get Freshwater Shrimps, or you can buy these at Wake's market stall during the Autumn Market.)
   
9. Fall Onion Rings
   
10. Ol' Pumpkin Pie (can be bought from Aroma Bakery in Autumn)

Once you have given him the Ol' Pumpkin Pie the Rabbit will show you to a secret plot of dirt where you can plan any kind of crop (no long dependable of the season).

Of course you can always wait until the winter market. If you have saved up enough coins you can buy the dishes at Sparrow's market stall. But where's the fun in that... (personally I only bought the Steamed Lobster there because I sold all my oranges at the end of the summer (!!) and would have had to wait until summer, year 2, to complete this secret).