2000 Cadillac DTS Clicking on start

I am wondering if my starter needs to be replace of is this a timing problem

When I try to start my car after sitting for a day or two I get a clicking noise as if the battery is dead but when I installed a new battery it starts up but if I leave it sitting for a few days it does the same thing, clicking as if the starter solenoid is not catching or if the starter is bad. What can this be? I have gone through a few batteries and I do not think it's the battery I installed a red cap battery recently and it was good for a few days and now it needs to be jumped and I am still getting the clicking noise and it will not start.