Actually, Boston weather in June is exceptionally pleasant. You'll expect temperatures, on average of under 70 degrees F. Boston is one of my favorite east coast cities. Since it's directly on the harbor leading to the Atlantic Ocean, the weather tends to remain fairly moderate. You can expect rain showers frequently.
I'd recommend you take a short and pleasant drive north on Interstate 93 to Interstate 95 to Kittery, just over the New Hampshire line, about an hour north of Boston, which has a solid mile of outlets stores with excellent bargains.
If you want to drive a little farther, Freeport, Maine is aobut 25 miles north of Portland and has the original L.L. Bean store and many, many other stores in a picturesque small town... (great lobster, too).
In Boston, be sure not to miss Filene's Basement, anyone can tell you how to get there. I'd never do Boston without visiting Fanueil Hall and everthing it has to offer...(It'll take you an hour just deciding what to eat!) See it here:
http://www.faneuilhallmarketplace.com/
Spend a few dollars and take the tour up Prudential Towers (about $12 for adults, less for kids) and walk
all the way around the Skywalk for a great view of the historic area of Boston...
...and welcome to the U.S., by the way!