starting problems

Hey guys - need a little help
I thought the reman starter was bad as when I turn the key to start position nothing would happen ... do it a couple of times and then the starter would drag and then start the car- so I put a New starter on and everything fine for a couple of days and now I have the same problem -
could the problem be the netrual safety switch?
I have done the following:
brand new starter - not a reman one
new heavy duty battery
cleaned all ground wires to make sure of good connections

thank you for any advice that ya'll can give me
Mike