The practice arena is Warrior Ice Arena - it's about 20 minutes west of the Garden. Virtually all their practices/morning skates are open to the public. But just as an FYI, I think they cordon off the section of seats where the players enter/exit the ice for "media only", and have a sign saying "no autographs, please", which makes sense - if the players sign for one kid, they have to sign for them all, because the one in back is always the saddest story. Adds an hour to their work every day.
But you're more than welcome to go to most practices at Warrior. During the playoffs, they MIGHT have either shut down access at Warrior, or they might have moved practices to the Garden - I'm not sure which.
There are two places to get info for practices. The first is the Bruins Week Ahead, on the website. They publish a news article similar to this one every Friday:
Boston Bruins Week Ahead: September 28 - October 4
If you click "News" and scroll down on the left side, you'll hit one of those articles eventually. The other place to check is the Bruins twitter account @NHLBruins. They usually don't ANNOUNCE practices or cancellations, but there's usually some activity on the account during practice.
Practice usually starts either at 10, 10:30, or 11, depending on schedules, and might go on for 60-90 minutes on-ice, I think. As a warning, they can get cancelled. You can park in the garage for free if you validate at Warrior - the machine is by the concession stand, and is free (no purchase necessary). It's not particularly handy to the MBTA rail lines, but is handy to the T's commuter rail - unfortunately, those run more on an "hourly schedule" rather than every 5-6 minutes during rush hour like the T.
Warrior does have its own website:
Warrior Ice Arena | Skate Where the Pros Skate I would urge caution if you look at their schedule - they block out extra time, so I'd go by the Bruins website and not the Warrior one.