Ah rugby, I dreaded it the first time I covered it too.
Scoring is sort of like football. If you touch the ball down in the end zone, it is five points (a try). If you kick a convert, it is an additional two points. A field goal is three points. You can score a field goal off a free kick penalty, or by drop kicking the ball through the uprights.
Basically, each team has a forward pack -which you'll see in the scrums and a back line, which is obviously out of it. The scrum half, who feeds the ball into the scrum is kind of like the quarterback running the offensive play.
Generally, possessions start with a scrum - the packs squaring off against each other and trying to tunnel the ball back to their own scrum half, a penalty play (the opposing team will have to go back 10 metres and it's a free ball), or a lineout, which is like a throw-in. Watch for steals off scrums and lineouts, that's a big part of the momentum.
Also, you'll have things called rucks where if someone is tackled, they'll post the ball on the ground. People must decide if they're in the ruck or not and join at the back. People in the ruck can't pick up a ball off the ground, people out of the ruck can. They must be standing. Generally, the person putting the ball on the ground is allowed one move to set the ball or to pop it up to a teammate.
Something called a maul is similar to a ruck, but you never drop the ball to the ground, you just keep forcing it ahead.
Most of the penalties in the game are based on positioning. If you're on offence, you can't pass the ball forward, and if you drop a pass forward, it's called a knock on. The other team will receive an advantage to try to play past the point of infraction, otherwise they'll get a lineout or scrum. If something is happening repeatedly, they might get a free penalty play. Playing the ball on the ground in a ruck is also a penalty. You also can't block (or obstruct) and that's a free-ball penalty.
Defensive penalties occur if you're offside - past the ball, generally, if you fail to wrap while tackling, or if there's a high tackle. Defensive players can also be penalized for playing balls on the ground.
A very serious infraction on either side may call for a yellow card, which sends a player to the "sin bin" in the opposite end zone, generally for 10 minutes or less at the referee's discretion. A red card ejects the player. They are not replaced.
Otherwise, there's a few strategic things to look for. If a team is taking its penalty kicks and booting the ball straight out, that's called kicking for touch. They do that because they get the ball back in a lineout, with field position.
Inside the field, there are several strategic things. Some teams go into contact quickly to cycle the ball quicker, others will try to evade contact by drawing defenders, then passing back to where they were. Another common ball movement strategy is the push or overlap, where a team will capitalize on having more bodies to the side of the field.
Really, it's a lot to take in, but in a scrambled way, those are the basics of the game.