I would recommend the train to New York, try and get seats on the left hand side so that you get the best views of the sea on the way down. The trains are comfortable and you are guaranteed a seat. I think you may be able to reserve seats now, last time I used it (March 2020) you couldn’t. The system in us stations is odd, they don’t announce the platform or let people board until the last minute, however a queue of those “in the know” usually forms ahead of the announcement. I’ve been told that if you use a porter (costs a few dollars) you can skip the mad dash to the train and grab good seats. If you can reserve seats now then this advice is obsolete.
In Boston I would recommend the Tea Party Ship, it’s obviously touristy but that doesn’t stop it being fun. The Science Museum in Cambridge is very good. The Duck Tour leaves from the Science Museum and is good fun and it’s a good overview of the city. I love both the Museum of Fine Art and the Isabella Stewart Gardiner Museum, the MFA is vast and impressive the Gardiner Museum is quirky and a more manageable size (if your teens are gallery tolerant). The USS Constitution is worth visiting, it’s free but you need to take your passport to get in. The museum there is good too. A cruise round Boston Harbour is fun to do and if you had longer a whale watching trip would be good. The aquarium is also very good.
A good way to get a feel for the city is to go up the Prudential Tower, the audio tour is very interesting. Just near there is the Maparium which is a really unusual but fascinating attraction.
Use either public transport (safe, easy to understand and efficient) or Uber to get around. Everyone in Boston uses the T (underground), you can get a Charlie Card from the machines in the stations, I think. Actual taxis are expensive and in my experience the drivers are rude and unhelpful. They’ve made it more difficult to get Ubers and Lyfts from the airport but I would still recommend them over taxis.
I think August is too early for Ice Hockey. I’ve been but don’t really enjoy it, I prefer basketball and baseball. A baseball game at Fenway is a possibility, also the tour of Fenway Park is really interesting. I think there is a museum in TD Gardens which would have ice hockey stuff.
You can’t go to Boston without eating Italian food in the North End. Go to Modern Pastries with the locals instead of Mike’s with the tourists for Cannoli and other delicious cakes. Beat the queue in Modern by sitting in to eat. The is also a branch of Mikes in the food hall at North Station that doesn’t have a queue.
August is hot, everywhere is air conditioned, of course but the T stations, in particular are close to unbearable, the trains are fine. Just be aware that you might not want to be walking around too much in the middle of the day.