Hi Steven- welcome to JP from the left coast. The only group I know of that has regular meeting times is at Ferris Wheels bike shop on South Street. Its a good group to learn some short road loops and meet people and ride. I'd say its beginner to intermediate level of riding. THey probably have the meeting times posted on their website, they meet at the shop.
There is also Boston Road Club which has some rides posted on their website. They offer some longer, more intense rides starting out in Newton or further out.
I'm sure there are more but thats a good start for you.
.. a little more is there are two google groups. The jpbike and Rozziebike. they are very good about keeping us posted as to what's happening in terms of bike initiatives and announcing organized rides..
Also, meetup.com has biking groups.
Are you a bike commuter? I ride from Forest Hills down to the seaport area almost every day.
You might also want to check out the Charles River Wheelmen rides that leave at 8:30 every Saturday from Nahanton Park in Newton (about a 25 min ride from the monument) there are plenty of people riding all year long, and they split it into speed and distance for racers and recreational cyclists.