Do matter and antimatter annihilate or release energy? Do matter and antimatter eliminate each other or release their equivalent energies? I'm almost certain it's the latter as mass can't be destroyed, but when speaking of the big bang it's said if there were equal amounts of both matter and antimatter there would be nothing left. I wonder how that can be true if they don't destroy each other. It may have something to do with how energy decays into matter.
 A: 
Do matter and antimatter annihilate or release energy?

They typically annihilate and release energy. Check out electron-positron annihilation, and low-energy  proton-antiproton annihilation. 

Image credit CSIRO, see The Big Bang & the Standard Model of the Universe

Do matter and antimatter eliminate each other or release their equivalent energies? I'm almost certain it's the latter as mass can't be destroyed

Energy can't be destroyed, mass is destroyed in annihilation. 

when speaking of the big bang it's said if there were equal amounts of both matter and antimatter there would be nothing left. 

There is another way to destroy particles, you can "melt" them. See this RHIC article. It talks about a quark-gluon soup, but IMHO it's best to think in terms of pea soup. There aren't any actual peas in pea soup. The peas got destroyed. If you can destroy some particles without using pair annihilation, you aren't limited to having equal numbers of electrons and positrons and protons and antiprotons. You can get into what's called a "stability tip". Think in terms of game play, where a small initial advantage is magnified, and one side wins.  
Note that while people talk about baryon asymmetry, there's also a matching lepton asymmetry. And don't forget that you can't have a stable universe if one side doesn't win. I think it's a bit like a game of mixed doubles in tennis myself, and that the words matter and antimatter are a bit of a convention rather than being something that's justified by particle properties.    
A: Your intuition is quite reasonable.  We observe (based on cosmic rays) that the universe is basically all matter, no antimatter.  At the high energy state of the big bang we would expect an even balance.  Cosmological theories need to explain the asymmetry in baryons vs antibaryons
A: Have a look at this Big Bang graph:
Annihilation means that when a particle hits an antiparticle, there exists a probability that they would both disappear and the energy turned into other particle antiparticles. It is the quantum numbers that cancel each other. At the elementary particle stage of the Big Bang, the energy is carried by elementary particles in a thermodynamic soup.

I.e., in  the interval between 10^-32 seconds and 1 microsecond the energy/masses state of the universe was a soup of quarks gluons and all the particles given in the elementary particle table, with their antiparticles. This soup was expanding which means that in general particles/antiparticles might meet and annihilate, or keep their separate existence for a time interval, annihilate into other particles and keep the soup going. 
When the protons are formed by 1microsecond, it is the time to ask why not an equal number of protons and antiprotons, which one would expect by statistical analysis. This is still a research question for particle physics , called CP violation, and for cosmologists.
