If you are pretty darned sure what school would be your first choice, then you should be looking for opportunities to have discussions there. One example is to ask your recruiting contact about prospective junior days, which are events the program sets up to bring in several prospects at a time.
While not every junior day invitee will end up getting an offer, to be included indicates mutual interest. If you ask and are not included, that would tell you something, too.
Either side can initiate the discussion, but it's kind of like waiting around for someone to tell you they like you before you ask for a date. If they do approach you, great, but if not and you just keep waiting, it may never happen. So you need to be prepared to take the initiative before too much time slips by.
If you find yourself on campus for a junior day, before you leave you need to approach the recruiting assistant and ask a question like, "What is your time table for discussing offers?" This is a good way to express interest and to see how he responds. Sometimes the reply will indicate a willingness to discuss things with you right then and there. Sometimes you'll get a response that puts you off, which may be an indication that whether you get an offer may depend on what someone else -- someone they rank above you -- does. In that case, you need to take the hint and actively pursue other options.
I would not do this unless you were ready to make a commitment if the offer met your needs. You're asking them to make a commitment to you, and if you're not ready to commit in return, you're likely to annoy somebody. On the other hand, if you need 40% and they offer 25%, it is OK to say that while you greatly appreciate their offer, your finances require that you look for a better situation, but that if the offer were 40% you'd be happy to commit right now.
If you've shaken hands while on campus, then a thank-you note confirming the deal and expressing your excitement about becoming a part of the team is in order. If you don't get what you want right away, a polite follow-up note that expresses your continued interest is in order, and regular follow-ups after that are also a good idea.
Before you force this issue, be sure you've done all your homework. What do you think of the school's academics? Its environment? Can you afford to go there? What can you observe of the baseball program and how it treats its players? Is there a genuine opportunity at your position, or are you going to sit the bench for four years? How does the program do with placing players in summer wood bat leagues, or with developing players' pro potential at your position? Is the head coach a jerk? If so, can you live with that?
Early commitments to me are no different from committing at the last minute. If you've done your homework, fine. But don't get stampeded into something if you don't really know what you're getting into.
