How do I ensure my wood screws have a minimum of 120 lbs pull out or shear strength?
The pull-out strength of a wood screw is going to be more dependent on the wood into which it is installed than on the material of the screw. The geometry of the screw thread also plays a role. "Cheap" screws therefore aren't what you need to worry about, instead, focus on the screw size.

Pull-out happens when you have a shear failure of the wood. A couple of formulas have been proposed to make an estimate of holding power

The Forest Products Lab of the US Department of Agriculture proposes the following:

p=15700*G^2*D*L


where p is the pull out force in pounds; G is the specific gravity of the wood (density of the wood divided by the density of water, see table 5-3 here); D is the shank diameter of the screw in inches; L is the penetration of the threaded part of the screw in inches. (Assumes reasonable pilot hole, screw into side-grain, etc...I haven't communicated all the nuance here. If your application is important enough to be calculating the pullout load, read the whole relevant section of the source document.)

Others have proposed a simple formula (stated incorrectly after the link, corrected here by me) related directly to the shear strength of the wood:

p=pi*D*L*S


where p is again the pullout strength; pi=3.14; D is the diameter of the screw, L is the length of engaged threads, and S is the shear strength of the wood. (make sure your shear strength is in units compatible with your length, e.g. if your length is in inches, your shear strength should be in PSI). 

This simpler formula idealizes things and assumes that all the load is carried by a cylindrical region of the wood that is the diameter and length of the embedded screw threads (the pi*D*L above is the area of this surface). If you use this formula, be sure to reduce the actual load by a generous safety factor of 2 or more.

Personally, I'd read the FPL report and use the first formula.