There are almost unlimited species of wood you could choose from for the project, but most folks settle on a few common choices.
For plywood, we recommend birch for its low cost. Baltic Birch or Russian Birch is more stable with a beautiful edge grain.
For your other lumber, pine is a common choice if you shop at a big hardware store like Home Depot or Lowe's. Look for the select/whitewood section for nicer quality.
A hardwood mill will have many more choices. Try Poplar, or ask for a "paint grade" wood if you'll be painting your project. Basswood is a common choice for stained projects, but ask your lumber dealer for their "stain grade" recommendation.
We have more details and suggestions for buying lumber in our plans.