These aren't all the rules, but they should be enough to get you started playing competitively. This is also a good thing for parents to read, especially #12. For players, it's a good idea to have your parents read these - especially #12.
1) You get two chances to get your serve in (referred to as “First Serve” and “Second Serve”). If you miss both, you lose that point.
2) If a serve hits the net and lands in, that is called a "let", and it's basically a do-over. It doesn't count as a missed serve - you just stop the point and do the serve over. It is the returning team's responsibility to call a "let".
3) For the rest of the point, if the ball hits the net, you keep playing the point as if nothing happened. The "let" rule only applies to serves.
4) When the ball lands out, call it "Out" loudly. If it lands in, don't say anything. If you say "in", your opponents may think that you called it out and then everyone gets confused.
5) Say the score loudly at the beginning of every point when you serve. If your opponent doesn't say it on their serve, ask them to tell you what it is. If you disagree with them, stop and try to figure it out before playing the next point.
6) When the score reaches 40-40, that is called "Deuce". In high school tennis, we play that the next point wins the game. When it is deuce, the server asks the returners which side (left or right) they want to return on.
7) JV matches play what is called a Pro-Set. The first team to win 8 games wins the match. But, you have to win by two, so if the score is 7-7 and you win the next game, it is 8-7. You haven't won yet. Play one more game to try to get to 9-7. If it gets to 8-8, call a coach over and they'll walk you through a tie-breaker.
8) You cannot touch the net during a point. If you touch it, you lose. You have to call it against yourself. You also must wait until the ball has come to your side of the net to hit it.
9) Players are responsible for making all the calls on their own side of the net. That includes almost everything:
Which shots are in or out
Whether the serve was a let or not
Whether the ball bounced twice before they hit it
Whether they touched the net
Whether they reached over the net.
If you disagree with your opponent's call, you can politely ask them, but it is their call. If you think they are making a lot of bad calls, you can call a coach over to watch over the match.
10) If a ball rolls onto the court during the middle of a point, any player can call a "let" and re-start the point. Coaches really want people to do this - the ball on the court is a safety risk and a distraction.
11) Wear clothing that will allow you to put a ball somewhere (in a pocket if you are wearing shorts, shoved up the leg of spandex if you are wearing a skirt or kilt). You should always have two balls when you go to serve, and you need a place to put the second ball. Setting it on the court next to you is a very bad idea - it's very easy to sprain or break your ankle by stepping on a ball.
12) Aside from cheering, parents and other spectators cannot be involved in the match in any way. They cannot offer advice, coaching, help with line calls, help with scoring disputes or anything else. Parents and spectators should cheer for their players and that is all. This is very hard for most parents - especially when you know they have the score wrong or you see a really bad line call from the other team. But you have to stay out of it. Even coaches will generally not get involved, except to over-rule their own team. For example, if one of my players makes a bad call, I will over-rule it. If the other team makes a bad call, I am not allowed to do that.
13) A ball is not out until it actually lands out. If you touch it before it lands, you lose the point. You also should work very hard to not call it out until it actually hits the court.
14) If a ball lands on the line, it is IN. If the ball lands 99% out, but 1% on the line, it is IN. If you are not sure if it was in or out, call it in. If you make a close call, your opponent might ask you if you are sure. Since you only call it out when you are sure that it is out, immediately say, "Yes, I'm sure" with confidence.