So, I'm hanging my tools in my garage, the wall is unfinished so I have direct access to studs. I'm using red utility hooks very much like the one pictured below to hang shovels, rakes, a few electric yard tools, etc. The screw end of the hook is probably between a quarter to half an inch wide, so drilling holes and screwing it in takes about 1/3 out of the stud material. Should I instead be installing cross beams between the vertical studs (Make something like an H between the studs) and screw the hook into that?
The hook is going to be inserted horizontally, so the load is mostly shear (downward) on the hook itself. You are in effect compressing the shaft of the hook downward on the bottom of the screw hole. There is some tension pull outward on the wood, but it is minimal for the types of items you listed. Studs made of 2x lumber can easily handle light loads in this configuration.
I would not try to hang items weighing several hundred pounds in this manner.
Also, I would not drill several holes, closely spaced, one above the other. While this would still probably hold your loads with little difficulty, it might compromise the integrity of the stud overall.
Drilling into the grain of a 2x4 stud might not hold up for very long. The wood will likely split over time, and the strength of the hook with diminish. If you must drill into a stud, first drill a pilot hole the size of the inner diameter of the screw on the end of the hook.
I would advise installing a cross beam between studs, like you suggested; you will get more use out of the hooks since the spacing can be customized and the wood will last longer. Still though, drill a pilot hole into the cross beams before you install the hook to prevent the wood from splitting.