You may use direct shear test to check the strength of the stabilized soil. 

You may have a problem collecting an undisturbed smple of sand soils unless they have significant clay content. I would go a different way. I would do CPTs prior to improvement and then additional CPTs after. Compare them to get the strength gain.

The test you should run depends on what you are doing.  How strong is the stabilzed soil?  What is the purpose of the stabilzation and how was it treated...

I agree with Jeffery fully. Before answering your question, some more information are needed. However the easiest way is to do as per Scott comment. CPT before and CPT after. This will be good if the depth is more than 30 cm.