The differences have been made fairly well above. Personally when building brackets, tabs, etc. in the shop I use cold rolled, on the thin stuff I use P & O steel (pickled & oiled) so it doesn't have the mill scale. This is especially important if I am going to weld on it, it just makes a nicer weld without all of the prep work. If I have a bracket that I am going to have to bend in the brake then I use hot rolled as the cold will generally brake when bent beyond about 30 degrees. Just remember if using hot rolled to remove the mill scale. I remove it prior to bending because it is much easier to remove than after the piece is bent and ready for welding.
As far as thicknesses it really depends on what the brackets are for and what kind of weight they will be carrying as well as how they can be supported or triangulated. I stock .125", .187", .250" and .375" in 1", 1.5", 2", 3" and 4" widths so I always have whatever I need with minimal cutting. I would say the ones I seem to have to replenish the most are the .125" x 1.5" and the .187" x 2". Mike.