You will need a hook knife, also called a crook knife or spoon knife to shape the bowl of the spoon.
The cheapest hook knives with decent quality are the Mora knives, but this one area where mass production does not necessarily do justice to the complex curved blade shape needed for a good hook knife. So the Mora family of hook knives may be good to get you going but this is one area I would absolutely recommend treating yourself to a more expensive knife in the medium term.
The Mora knives come in a range of curves to the hook and with both single and double edged blades. the Morakniv 163 is a gently curved and double edged blade. The advantages of this design are that you can cut to the right and left without needing to change tools, but with only a small part of the blade near the handle available to press your thumb against, this knife does take some getting used to. The Morakniv 162 has a tighter curve on the blade and is again double edged. it is of course possible to grind off one of the edges to create a standard single bladed hook knife
The Morakniv 164 is similarly more tightly curved is a single edged blade which makes it more controllable. It comes in a right and left handed version.
A Morakniv 163 double bladed hook knife with home made leather sheaf
A pair of left and right handed Morakniv 164s