how to install tech annilation or any other mods?

Join tech anihilaiton room, and zero-k lobby should auto-download whatever is needed.

Uncheck the "official only" box on the multiplayer server list, and then lots of rooms for other Spring mods will appear. They can then be joined normally.

Dunno if you can host room with tech annihilation with zero-k lobby. But there must be some empty rooms if you wanna just try it out with friend or bots.

Or you can just launch spring.exe directly beware to have correct version of engine maps and so on, but this will work only for singleplayer tryouts.

Anyway I really dont recommend creating rooms other than ZK since you are newb, its not very easy, but if you insist, I think springlobby would suit you best.

That doesn't seem to be correct, Tech Annihilation does not appear in Host Battle menu in ZKlobby

To host it You need write its version name on it