A combination of wood screws and glue actually makes the strongest cabinets and is better for cabinets holding heavy objects, such as those in a garage. When using screws, you will likely want to also use clamps to keep the cabinets square as the glue dries.
For most indoor wood projects, plain yellow wood glue is still used most of the time. It forms a strong bond and provides about 15 minutes of working time to get all the parts clamped properly and square. It is easy to apply and cleans up easily from your hands and tools.
For joints inside of the cabinets that will not be seen or perhaps don’t fit as snugly as you would have liked, consider using a urethane glue, such as Gorilla Glue. It provides plenty of working time until it starts to set and then it expands. This fills up any unwanted gaps. It is not quite as strong as dedicated wood glue, but the Gorilla Glue is waterproof, whereas wood glue is not.
There are quite a few options for clamps, and for most cabinet projects, you will need several different types. For many of the smaller, light-duty tasks, simple spring clamps and sliding-arm bar clamps are adequate. They apply only a moderate amount of force to the joint, so the pieces should fit properly before gluing them.
For projects that have a large circumference and for which you need to hold several pieces together, a flexible band clamp is ideal. The fabric band can fit around irregular shapes, and the hand toggle draws them together. As with the above clamps, the force is not strong, so it cannot force pieces into proper orientation.
Standard C-clamps are inexpensive and can provide a high amount of clamping force. The force is distributed over a very small area, so it can easily mar the surface of the wood. Unless the wood surface will be hidden from view, always use clamping pads to protect the wood surface.
Hand screws are another type of clamp that can apply a substantial amount of clamping pressure. With its long wood jaws, it can distribute the force over a large area without marring the surface. It can be a little tricky learning to coordinate the tightening of the two screw handles, so practice. Hand screws also provide some control over the alignment of the joints.
For large jobs, such as your cabinets, heavy-duty bar or pipe clamps can be used to force the pieces into their proper alignment. With these types of powerful clamps, it is wise to use cauls to protect the wood surface and help direct the alignment force in the proper directions. Cauls made from melamine tend to release easily from glued surfaces.