Start with the basics:
1) The object is to score the most points. There is a time limit (unless there is a tie when the time runs out, then there are special rules to address that) to the game.
2) Each time a team has possession of the ball, they get 4 tries to move the ball a minimum of 10 yards towards their goal (to score). If they succeed at any point during those 4 tries (either go 10 yards total forward progress, or further), then they get ANOTHER chance to try it again. This is called getting a "first down". So, if they get the ball and only go forward 2 yards, they refer to it as 2nd (2nd try, so "2nd down") and 8 (which means they are on their 2nd try and have 8 more yards to go or they lose possession of the ball).
3) If a team does NOT move the ball 10 yards forward during their 4 tries (downs), then the ball is turned over to the other team AT THAT POINT ON THE FIELD. So, what typically happens is that on 4th down, (their last try before turning the ball over to the other team then and there) they will punt the ball to the other team. Yes, they are essentially giving up their turn, but they get to make the other team start THEIR turn a lot further down the field (as far as their punter can kick it). Sometimes, when they are only inches from reaching their 10 yards and a new first down, they will opt to "go for it" and there will always be lots of talk by the commentators about this move. And fans watching the game. Because it can be very bold and smart, or very risky and stupid, depending on if they get the first down or not.
Once you understand those basics, you can add bits of knowledge as you go. There are so many special rules for special situations that I just ask as they crop up and learn from year to year. :)
Overtime rules were fun for me this year in the Giants last playoff game, lol.
Here's what you need to know about penalties: When a ref throws a yellow flag, it is a penalty. If they blow the whistle and throw the flag before the ball is snapped, it is ALMOST ALWAYS on the offensive team (the team with the ball). Different broken rules award different penalties. So holding might be 10 yards (the offensive team starts their down over again further away than they were and they have to gain back that yardage to earn a first down), pass interference is awarded at the spot of the foul (if I remember right), so this can be a doozy. Personal foul can range from 5-15 yards (a facemask is always 15 yards).
These can make it nearly impossible for a team to move the ball down the field towards their target: scoring, or getting the next first down, so that they can continue to get more turns.
Also, there are rules about certain things that can or can't be done-- a quarterback has rules that govern his field position when he throws a pass. He can only throw it forward down the field if he is behind the line of scrimmage (where his team lined up on the ball at the start of the play). If he has run out from behind his linemen (due to pressure from the opposing team getting past them) and runs forward beyond the line of scrimmage (if I understand this correctly) he can ONLY throw the ball sideways or behind him (a lateral pass) or a flag will be thrown and a penalty given.
If a quarterback is sacked (tackled with the ball) in his own endzone, the opposing team earns 2 points. If a foul/penalty is called on the offense when the quarterback is in the endzone with the ball, it is also 2 pts for the opposing team. These are called getting a safety.
So... your team always needs to move the ball a minimum of 10 yards forward within 3 plays (or they will kick it away). And they need to avoid penalties and protect their quarterback. These things will go a long way.
If your team can throw (has good receivers and good line protection for a quarterback with a decent throwing arm) AND run (good running backs that can gain yardage running it not throwing it) it is a really good thing. Not just because it gives you more ways to effectively move the ball and your one good guy doesn't get worn out. But also for the strategy involved: the other team has to "protect" against ALL OPTIONS you might use. That's why the get in the huddle to tell the next play to their teammates. The other team can't know what they plan to do or they can defend against it better.
There is a lot more, but this is just a good, general basic outline.
Hope it helps you enjoy Sundays game!
Go Big Blue!