What kind of box do you want to make? The accuracy and straightness of the cuts becomes very important when making boxes. Deviations in one side become nightmares on the opposite side. If you wish to make what they call "band saw boxes the smallest blade size will allow the best curves, but bandsaw boxes require a lot of sanding to remove the tooth marks in the cuts..
I have made a lot of boxes, mostly nice accurate tightly joined designs and all cube shaped in varying sizes. The kind of box you want to make determines the kind and size of tools needed.
If making what is called "Bandsaw Boxes", you will need some idea about the largest and smallest size you want to make, as the bandsaw size is determined by the size of your project size range.
Most little whimsical band sawn boxes involve cutting the outline iof the desired box n a thick piece of wood. The next step involved cutting a thin layer off the thick piece of wood. It will become the back of the cut out box to keep the drawers from pushing through after you cut and remove each of the drawer blanks. You should also cut and temporarily remove a thin slice off the face side of the work and then temporarily glue it back on. Something like double sided tape will hold it in place..Then with the facing side reattached, mark and cut the shapes of each drawer in the workpiece. and keep track of each drawer face piece as you cut them free. It helps to number the piece as well as the core. Once they have been cut and removed, you can then glue the back piece back on. If there are any irregularities in the cut when you cut off the back, gluing it back on in it's original orientation will make the seam almost invisible. Then you can cut the inside shape of each box blank, both sides and bottom must remain uncut and only the center piece and top removed. from each of the box blanks.
Now it's reassembly time. Glue the drawer facs to the mating box sides and bottom. You will need to make thin backs or inserts to close up the back end of the drawers. You can un-attach the box face frame piece, since it now has holes in it for the drawers, and using care, glue it back on with permanent glue as you did with the back.
Expect considerable sanding at this point, but you can't remove too much or the drawers will fit too loose. Be sure to match the drawer face to the correct drawer box sides/bottoms. The drawers won't have backs, so making some thin backs of the outside profile of the drawer sides and bottom will be needed. You can then glue each of them together. If each is a slightly different size/shape it will be easier to keep the correct pieces together.
You will likely cut several boxes before you master the cutting and assembly process. If I remember correctly, I made three before I made on that I was happy with. Small/tiny versions can becut on a scroll saw, but considerable patience is needed both in cutting and assembly. Large versions may exceed the capability of the saw size, so a larger saw would do the job better. Tight curves require smaller blades. To cut out the drawers from the blank, you will need to make a single cut from one drawer hole to the next that gets later glued back together. Careful planning is required to position these cuts where the lost wood from the cuts will still allow a tight joint at re-assembly.
There is a lot of complexity involved in making small bandsaw boxes. You will get better with experience. I haven't made on in over 50 years, so I would likely need to start over in the learning cycle.
Charley