Since you were thinking about commuter rail from Lawrence, I'll assume that you have good access to I93. If so, I would suggest taking I93 to Rte 16 in Medford. From there, you are less than 5 minutes to Wellington Station on the T Orange line. Its 3 stops to North Station and the Garden, 4 stops to Haymarket and the North End. Fare is purchased at a vending machine using cash or credit/debit. It's a good idea to buy your round trip on the way in, because there can be quite a line at the fare machines after the game.
Once you get there, bear in mind that the gates don't open until an hour prior to game time. In addition to the local stuff others have already noted, there is a very large shop at the North Station level with plenty of Celtics and Bruins stuff for sale. Don't buy more than you can carry in a small bag. No backpacks, packages etc allowed in the Garden. Also, once you get in to the Garden upstairs, they have more shops, plus displays of the original parquet and some other stuff. They frequently have a table where they hand out green and white balloons for the kids, and you may see Lucky signing autographs and visiting with the kids.
The balcony is a lot of fun. Lots of serious fans and characters to watch. And your kids may learn a few new words.
Don't be surprised if the long day and excitement takes a toll on the kids. It was never an issue for the Fat Kid, who always stays til the bitter end, but some kids get restless/cranky/tired and you might have to make an early exit. If so, the Orange Line trains run every 10-12 minutes, and you are back at the car at Wellington in no time.
You and your family are in for a great time. Enjoy!