Sounds like one of two things. First thing I would check are the wire connections. Make sure that they are all clean and tight, particularly the grounds. If you have a three post solenoid make sure that the base is fastened to a clean part of the frame, that is the ground. If that doesn't do it test the starter, by jumping it right at the starter, bypassing the key and solenoid. If it spins easily you will have to go back to the wiring. You said that you replaced the battery, but you would not be the first one to receive and bad or under sized battery. I recommend a battery with a minimum of 375 CCA for all of today's units.
