Correct Screw length I should be buying

Just add up all the parts:

• Bracket - your questions says "about 1 inches" but I usually find these types of brackets (but haven't used exactly this one) are a bit thinner than 1 inch
• Cabinet back - varies, might not be any at all
• Drywall or other wall material - typically 1/2"
• Between 1 inch and 2 inches into the stud. Studs are nominally 3-1/2" thick, but if there is any possibility of wires going through the studs then you don't want to go more than 1-1/2" in.

So that makes a total of 2-1/2" to 3" typical.

As noted elsewhere, don't use drywall screws. They are handy, but they are not designed to hold a heavy load, which is important for shelving brackets.