When the group first enters, and you have new members, give out slips of paper and pens and ask each person to write three (anonymous) random facts about themselves (you could ask for specific answers e.g. How many pets do you have? What is your favourite music band? What is your favourite ice cream? Where was your most exciting holiday?). Now mix the slips in a hat and hand them out again. Each person reads theirs out and either tries to guess who it refers to, or asks the person who wrote it to stand up. S/he then introduces him/herself. You as leader could then ask them to expand on their answers. If you plan an activity that needs pairs next, you could ask pairs who emerge from this to stay together for the next activity.
Get everyone sitting in a circle on the ground (you could go boy/girl/boy/girl to integrate them more). Each person must put their hands palm down on the floor in front of him or her, away from his or her body. Then each person must bridge their left arm over the right arm of the person next to them so everyone's arms are crossed but still touching the floor. We now have a circle of hands on the floor. The leader taps the ground once. The hand next to his on the left also taps the ground once. Keep this going round the circle. The game is, of course, tricky because having arms crossed confuses each person as to when their turn is. To make things more difficult, at any point a person can tap the ground twice and change the direction of the game. Try speeding things up too. If someone even flinches a hand when s/he is not meant to, then they must remove that hand. If they mess up again, then they remove the other hand and are out of the game. Hopefully, this should cause many laughs and break the ice.
Ask your group to move into line with those born in January first, and so on through to December. Now ask the smallest lightest member to stand on a table at the far end of the room. The group should now line up in two groups (January - June, July to December) facing each other and leading away from the table. Get them to link arms with the person opposite. This creates a 'human hammock' for the person on the table to fall back onto (they should face away from the group or be blindfolded). Yes, this is a game of trust and should get the group working together. You could keep them in 'January' groups, 'February' groups and so on for the first couple of sessions whenever you do Bible studies or small-group games.