Our method of staining is as follows
1 coat of Mohawk brand wood dye to set a base layer color. Lightly sand to add color variation/distressed look
1 coat of stain, let dry over night.
2 coats of clear shellac. Polish by hand with a rag. You can also leave out the wood dye but you may need more layers of stain to get an even color. Stain won't soak into highly sanded areas as well as rougher areas so it will appear lighter. Dye penetrates pretty evenly no matter how much or little the area is sanded, that's why I use it so set a uniform base color before staining.
Otherwise here's an informative article on Baltic Birch that should get you started. https://www.woodworkerssource.com/blog/woodworking-101/tips-tricks/your-ultimate-guide-to-baltic-birch-plywood-why-its-better-when-to-use-it/