My battery is very weak?

My car is making a weird dinging sound when I start it, like my battery is very weak and the other day it died out completely. The other day I noticed every time I get in the car my radio is reset, then my trunk started popping open by itself while I was driving and then my car died out completely while it was parked and when I went to restart it all the lights were out, no radio, it wouldn't even see n turn over so I went to use y remote to pop the trunk and as soon as I hit the trunk button and the unlock, the car started back up. Then, this morning when I turned the car on, it made that same "dinging" noise w/customer sounds like the battery is very weak and then all my brake lights went on in the dash and my clock was reset again.

Then I turned the car off at the gas station and it wouldn't restart or turn over again and after a few minutes it started.

I know we enough that it sounds like my battery or alternator issue but what confuses me is the trunk popping open and the fact after I hit the remote button the car started again. If it was my alternator or battery, I would think it wouldn't just die and then start again. I am soooo confused! Could it be my electrical system or a short somewhere? How do I even go about fixing this or finding out what's wrong without going broke in the process? I have very little funds available to me..