My 2008 Mazda 3 is having electrical problems

Recently I got a new battery for my Mazda 3, and after I did that every time I turn my headlights on my stereo turns off. I don't know if they're directly related but it was kind of coincidental that after I replace the battery that started to happen. Also now every time I start my car for about 3 to 5 minutes there is a ringing sound a steady ringing sound that comes from the battery area. Now I've read of similar problems people were having with their Mazdas such as my car just wouldn't start wouldn't turn over the gauges were going up and down like crazy I was able to jump start it and get it moving again but once I got to where I was going and I tried to leave I reversed my car and then the car just turned off wouldn't turn over wouldn't do nothing and then a couple seconds later it started again but then it just immediately died and then wouldn't turn over again and then that was it it wouldn't start again wouldn't do nothing. I need some advice please people