Welcome, JonasWallace!
The first thing that you have to realize about trying to understand this game is that you have to take everything you know about previous A&A games and chuck it out the window. When you read the rules, treat this game as though it were not an A&A game at all. Any preconceptions you bring from other A&A games are just going to confuse things, as you are going to read things into the rules that just aren’t there.
That being said, when units attack other units, they only fire once per combat step (Air, Sea and Land) in which they have an attack value. In your example, your land units would not get to fire at all during the Attack Air Units step, since none of them have air unit attack capability. However, when his fighter fires in the Attack Land Units step, it will fire only once, and then combat is over for that step. His fighter will attack your land units without any response from them, but it will only roll once.
Also, the combat steps are each performed once per turn, not repeatedly. After each step (Air, Sea and Land) is done one time for each zone, the combat phase is over for the turn.
Does this help?