Give It Time
Don't expect to be best pals with people in the group right from the beginning. Every group has its own personality, and as a result you need to figure out what makes it tick. This is true for groups of casual friends and for those in a more formalized group, like book clubs or sports teams.
Groups may have people coming and going a lot, so if they don't warm up to you right away, don't take it personally. They may initially assume that you're going to be yet another person who signs up for their group and then quits. When group members know they can trust you as someone who wants to be a part of their circle, acceptance will be easier.
Show Up to Group Activities and Don't Rock the Boat
In the beginning, go with the flow, even if it feels odd to you. Trying to suggest new ways to do things right from the beginning may rock the boat too much. Instead, get to know the individuals and how they rank and interact. After that, you'll have an easier time relaxing and showing your true personality.
Be good to your word and show up to the things you say you will. Let your actions speak for your character.
Get to Know One Person Who Can Help You Get AcclimatedA good strategy for getting acceptance into a larger group is to "divide and conquer." You'll probably feel closer to one or two people immediately, so give some time and attention to the individual friendships that develop. Don't use these friendships to undermine the group or start a gossip train, but instead allow one or two people to really get to know you. They'll be more willing to include you in things and talk you up to the other members of the group this way.
Ask Group Members About Themselves
People like to tell their own stories, so get to know the folks in the group by asking them to tell you about their kids, spouses, jobs, hobbies, and life. A great question to start with is, "How do you know the rest of the members of the group?" or "How did you first come to join this group?"
Listen closely, and see what you both might have in common. If you're currently new in town or recently out of a relationship, for example, you might notice that many of the group members were once in your shoes as well. Tell them bits of your personal life in an effort to connect.
Don't Get Discouraged
Getting to know new friends (no matter if you do it individually or in a group) is hard work. Conversations take more time, you need to be on your most pleasant behavior, and you'll feel as if you're doing everything "their" way in the beginning. That's okay. In fact, it's normal.
But while all this is going on, don't give up. It might not feel good or comfortable in the very beginning, but it will get easier as time goes on. More than that, it's worth sticking in there to see if the group has potential to introduce you to some good friends.