Unless the garage is part of your heated/cooled space, the basement is probably better.
If you have exposed joists, a simple system is a set of "U" brackets (typically of wood) hung from the joists, which keeps the wood up off the floor, and away from the walls if those are prone to dampness. If you have a tall basement and are not overly tall it can be clear of your head height, or you can build it deeper and walk around it (put a bench or something under it so you are less prone to walking through it/lifting your head into it.) Or, sometimes you can screw a crosspiece directly to the bottoms of the joists, and store lumber on top of them, between the joists. You do need enough room to get the lumber in and out, lengthwise, for these methods. Most of the credit for this would go to my late grandfather.
Finished ceilings make it more difficult, but not impossible.
Or you can go freestanding and put a couple of sections of adjustable wire "commercial" shelving in line, and use one or more levels of that for wood storage, and the rest for other storage.