My Mobo is gigabyte ga-h97 gaming 3.
I'm thinking of getting NVIDIA GeForce GTX 760 but not sure If I should get AMD Radeon™ R9 270X instead.
I think the monitors I have only work on vga since it doesn't have any other ports. I tried checking online but was not able to find any information about them so far.
My monitors are samsung s20d300f and s20d300. Please tell me if you have information on these as well.
Your choices don't seem to line up with the budget you said you have. The 270x is currently $144 on Amazon. A new card closer to your $260 budget will obviously be much stronger than a $144 GPU.
Neither the 270x or 760 make sense if you have a $260 budget. The r9 380 that was recommended before makes much more sense. If you prefer Nvidia the 960 makes much more sense than the 760.