Q: My vehicle won't start

There's alot of corrosion around the battery wires, the lights work but it won't turn over. Someone told me that it's the alternator. Could it be the corrosion that is preventing the car to start? I've done my research and i am pretty knowledgeable when it comes to cars because of my dad. But being as to I'm a female nobody will listen. I'd rather start small before jumping to the big repairs. It sounds like it wants to turn over but it don't. There is a clicking noise in the relay box when attempting to start it. What is the easiest way to get rid of corrosion? From what I know baking soda water and a wire brush? Please give me a little feedback. I also listened for the fuel pump and it does have the proper noise it should have when power is going to it (I checked that because of the clicking in the relay box)
My car has an automatic transmission.