Yes, they do sell out pretty quickly. They released some tix a few weeks ago. Last year when they did I was able to get one of the 4 game packages within an hour; this year I spent several hours in a virtual waiting room and when I finally made it into the purchase area, I got booted off. I did manage to get three games later, though, but it took almost all day.
If you're not on the redsox.com email list, get on, they notify you of events (like ticket sales). Keep watching every day, eventually they're going to release more tickets for summer games. Kind of play with it now to get an idea of what type of seats you want and where, what your price range is, etc. If worse comes to worse and you can't get any, I was told by a ticket agent at Fenway last year to call on day of game, you can often buy standing room tickets the same day. I also know that some people have had luck going to the box office the same day, too (of course, you'll have best luck on days that unpopular teams are in town).
Lastly, sign your child up for Red Sox Kids Nation - one of the benefits is that they set aside tix for RSKN members and you might be able to get some that way. (I've never actually used this benefit but I know it exists.) A fun thing to do when the team's out of town is to take the tour of Fenway. It's offered daily at 4pm, tickets sold across the street at the gift shop.
If I hear of anything I'll PM you. Good luck.
Huge Yankee fan here. There are online ticket brokers, and also a place where ticket holders sell - stubhub.com
