If you just bought a manual miter box, a simple modification will extend its useful service life. Insert and fasten a piece of 1x stock to the inside bottom of the box. The saw blade will then cut into the false bottom, without damaging the miter box itself. Occasionally move the wood back and forth, or turn it over, to spread out the wear. When badly cut up, just replace the piece.
If you’re just starting out, three power tools for general woodworking that are worth spending more money on include a good tablesaw, router and drill press. With these three, plus some hand tools, you will be able to build many of the projects in how-to magazines. Which tool you buy first can depend on the projects you are planning, but a good saw usually gets priority.