2002 Mercury Sable starter will not start the car

I just replaced the starter in my car. When I turned the key to the on position, I hear a click and the lights in the dash board flashed. I have charged the battery fully and also jumpered the battery using my other car and I get the same result.
I have checked the cables and I had someone else check the cables and they seem ok.
Could a bad starter cause this issue?
Also, the radio does not work anymore, so I think that a fuse was blown, I will check the fuses later today.
Do you have any recommendations on how to proceed?