There is a multitude of tests for FRP composite materials. They could be categorized for material (resin and fibers) characteristics, mechanical properties, durability and long-term characteristics.
The ACI 440.3R-12 provides a list of some tests and those transferred to ASTM. (Check the attached Table 1.3). ASTM Volume 15.03 includes most of the tests that you may be interested in. Please check the following link for a list:
The following are also some of the most commonly used ASTM ones for FRP composites as well:
ASTM D792. “Standard Test Methods for Density and Specific Gravity (Relative Density) of Plastics by Displacement.”
ASTM D4475. “Standard Test Method for Apparent Horizontal Shear Strength of Pultruded Reinforced Plastic Rods by the Short Beam Method.”
ASTM D5028. “Standard Test Methods for Curing Properties of Pultrusion Resin by Thermal Analysis.”
ASTM D4476. “Standard Test Method for Flexural Properties of Fiber Reinforced Pultruded Plastic Rods.”
ASTM D570. “Water Absorption of Plastics.”
ASTM D3171. “Standard Test Methods for Constituent Content of Composite Materials.”
ASTM D7205. “Standard Test Methods for Tensile Properties of Fiber-Reinforced Polymer Matrix Composite Bars.”
ASTM D7617. “Standard Test Method for Transverse Shear Strength of Fiber-Reinforced Polymer Matrix Composite Bars.”
ASTM E831. “Standard Test Methods for Linear Thermal Expansion of Solids Materials by Thermo-Mechanical Analysis.”
ASTM D3418. “Standard Test Method for Transition Temperatures and Enthalpies of Fusion and Crystallization of Polymers by Differential Scanning Calorimetry.”
ASTM D7705. “Standard Test Method for Alkali Resistance of Fiber Reinforced Polymer (FRP) Matrix Composite Bars used in Concrete Construction.”
ASTM D2344. “Standard Test Method for Short-Beam Strength of Polymer Matrix Composite Materials and Their Laminates.”
ASTM D3039. “Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials.”
Depending on WHICH PROPERTY you want to measure, but I believe that the ASTM standards are quite simple compared to the ISO which are a bit more complicated
The Department of Defense in US has published a fibre reinforced polymer matrix composite handbook, which contains all information about the various testing processes, its ASTM standards and prerecorded values for fibre reinforced polymers used for their defense applications. It can be downloaded through the following link.
The following article gives you the detailed discussion about the testing of FRP materials
Sivaganesh Selvaraj and Mahendrakumar Madhavan. (2018)."Strengthening of Laterally Restrained Steel Beams Subjected to Flexural loading using Low modulus CFRP: Experimental Assessment,"Journal of Performance of Constructed Facilities (ASCE) . DOI : 10.1061/(ASCE)CF.1943-5509.0001293.