I went to start my car, it clicked once and then nothing at all, it won't even click anymore. There is a blown 10 amp fuse under the hood in that fuse box. Replaced blown fuse and it again clicked once and now will not even click again. Only the red oil light on my dashboard lights up when I turn the key now, brand new battery. Any ideas of what it could be??

 

Forgot to mention its a 2002 seabring Lxi 2.7 v6

 

Check all the wiring from the battery to the starter motor. You are looking for corrosion which could be interfering with the flow of current. Loosen any connection you can, wiggle the wires, re-tighten.
Check for codes.
What was that fuse for that you replaced? There is a legend describing the fuses on the underside of the cover.

Also check the starter relay.

 

I didn't notice any corrosion, the fuse that was blown was for "starter/fuel". I had the alternator replaced last week is it possible that it caused maybe the solenoid or ignition to blow?

 

A blown fuse is usually (not always) the result of too much current being pulled through it. That, in turn, can be from a short or other circuit failure. A failure in a circuit labeled "starter/fuel" is an obvious starting point to look in a car that won't start. Something caused that fuse to blow. You need to find it and fix it. Chances are pretty good that when you do, the car will start again.