If I'm looking for a type of wood that has very little flexibility (or one that has a lot), what wood metrics should I look for in tables / charts / lists to judge relative flexibility?
Some background: I am a structural engineer by profession, have taken structural wood design classes in college, and wrote my graduate thesis on a topic of structural wood design.
As @ASTPace said, the stiffness of a structural member is dependent on both its modulus of elasticity (Young's modulus) and its moment of inertia (second moment of area). I have no idea of your level of engineering knowledge, so I'll explain a little about each below.
Modulus of Elasticity
This is a measurement of a material's resistance to strain given an applied stress. Strain is a proportionate displacement. For example, if a member stretches to 110% of its original length under load, its strain is 0.10. Most structural material exhibit strains much less than this before failure, so this is just a numerical example to illustrate.
Modulus of elasticity, E, is usually found by stretching a test coupon in an apparatus and charting the coupon's displacement vs. the applied load. For example, the image below shows the stress-strain curve for mild steel. The initial linear segment is what's used to determine the modulus of elasticity since that is the elastic deformation of the coupon. The modulus of elasticity is simply the slope of the initial segment. The stiffer the material, the steeper the line.
Moment of Inertia
This is a property of the cross-section of the structural shape and is independent of the material (assuming the material is essentially isotropic, or the appropriate corrections have been made for anisotropy).
For simple rectangular sections, the moment of inertia, I, is found as bt3/12, where b is the width and t is the thickness. As you can see, a change in the thickness has a much larger impact on the moment of inertia than a change in the width since the change is cubed. For example, doubling the thickness increases the moment of inertia eightfold.
Considerations for Wood Design
In reality, the stiffness of a material is measured by the product of E and I. Consider the image below, taken from the AISC Steel Construction Manual, 7th Edition. The equation for delta (deflection) is a function of 1/EI. As the stiffness increases (product of EI gets larger), deflection decreases.
If you truly want to compare two woods, you have to isolate one of the variables. For example, to determine which wood is stiffer, you will want to test two samples of the same cross-section. The stiffer one will be the one that deflects less.
The opposite is also true. If you have a wood with a lower modulus of elasticity, you can create a member of equivalent stiffness by changing its moment of inertia. For example, if Wood #1 has half of the E that Wood #2 has, Wood #1 must have twice the I of Wood #2 to be of an equivalent stiffness (so that the product of EI for each is the same).