We live the baseball life as well.
Like others, my DS obsessively hit balls from early on (he almost detached my mother's retina when he was 3yo with a whiffle ball she was pitching to him that he line drived into her eye,

she now has one pupil permanently bigger than the other

but she lovingly forgives him and insists "he didn't mean it", lol).
He did other sports but wanted to do baseball more than anything else. He took winter classes every year from the time he was 6 (which were invaluable to learning fundamentals like how to stand to hit and throw and how to throw safely so as not to injure your arm, etc) and every spring worked his way up all the rec levels, just having fun with it. His coaches always put him on the All Star teams, in some cases even before their own kids (which I was surprised at, but respected), and they encouraged us to have him try out for travel baseball (Cal Ripkin league

), which he eventually did and he really loved it.
One day when he was 9yo we went in a baseball training facility looking for classes and the owner asked if he could put him through the paces to "see what he had". He liked what he saw and asked if we'd be interested in his playing AAU baseball. We had never heard of it and when we looked into it, realized it was a huge commitment and neither us nor DS was ready to have weekends completely devoted to baseball at that point. But it planted the seed and two years later he decided he wanted to play, tried out for a different team and has been playing AAU ball ever since. Training is just about year round and as intense as Mickeyluver37 described. So he plays on Rec, CR and AAU teams. He will train as an umpire this year as well. He is still in middle school so we're taking each year as it comes. He'll continue to play as long as he's enjoying it (and as long as he devotes as much to his schoolwork as he gives to the sport!).
With that said, I'd like to say that you've gotten some good advice and a lot of different perspectives here. (I always get confused reading threads like these because different leagues are called different things all over the place!) I'll add a few things that we've learned not just for you, but for others reading who might also be interested.
There are a lot of good books on the fundamentals of baseball, often written by coaches and players, that are kid friendly. (DVDs and
youtube videos, too.) I had to buy many of them because it was one of the things that my DS always enjoyed reading (no surprise there, right??). Some of these books were written decades ago but aside from a few things that may have changed, most of the fundamentals are still the same.
There is more to the game than hitting and catching, especially as they get older, and even beginning around ages 9-10. Players should understand where to stand, what base to cover, who to throw to, how to use a cut off, how to base run, how to stay safe, etc (although coaches will direct players as well, of course - but no kid wants to be embarrassed out there during the heat of the moment). It can be amazingly complicated for a game that, by appearances, looks fairly simple. (I was captain of my softball team in HS and I've learned things I never even knew just watching these kids play, reading through the books with my DS, listening to him talk, etc). Fortunately, the kids get it pretty well when they put the time into it (as pp said).
Can a kid get by playing in Rec league without knowing this stuff? Sure. But (again, as pp's said), it does get a little harder as they get older becase on some level, there is an expectation that players will learn and at least some of the rules of play are going to be followed as opposed to just wildly throwing the ball around and fooling around, etc. In our town, in order to play CR in the summer, a player must play Rec as well. So Rec players will be with players of all levels and abilities. I have heard some coaches express worry that kids that aren't paying attention or don't have strong skill levels could get hurt with balls hit or thrown by players more used to playing advanced ball. And by 11 or 12, some of these kids are enormous, so their throwing and hitting power can be pretty intimidating (and dangerous).
The other thing I'd just like to add that I haven't seen here are some of the "other" things that make a good player and a player a standout. And that is
paying attention to the game. DS had one class early on that the coach would randomly call on players to tell what inning it was and how many outs there were, etc. Players who could not answer were

It taught DS the value of knowing what's going on at all times, whether he's on the bench, at bat or on the field. Ted Williams talks about that in his book
The Science of Hitting. Part of being a great hitter is not just how you swing the bat, but studying and challenging the pitcher and other things you wouldn't think of. So, not paying attention and/or fooling around on the bench signify to coaches that a player is not really serious about the game, and that is the player that may not see as much playing time as some of the other kids, truth be told (from what I've observed on various teams). No coach wants to waste their time. So stress to younger players the value of learning the game and paying attention.
Of course in hitting and fielding, as a pp said, repetition, repetition, repetition. Fielding grounders is important. Fundamentals like keeping two hands on the ball and putting the glove to the ground are something that young players should practice over and over. And a good warmup is critical to preventing injuries at each playing session.
Looking forward to a fun upcoming season here and wishing the same for all of you as well.