The Icelandic Red Cross (Rauði krossinn, say: ROY-thi KROSS-inn) is the easiest first door. It runs free social activities in many towns, not only help for people in crisis. Depending on the branch, that can include language cafés, open houses where anyone can come for coffee, a visiting-friend service for people who are alone, and groups for young people and for newcomers.
You do not need to be poor, in trouble, or a member of anything. You also do not need good Icelandic. Volunteers are used to speaking slowly and in English.
Check what your branch runs: Icelandic Red Cross (English). Branches are listed by area, and each one publishes its own weekly activities. If nothing near you is listed, write to the branch and ask. Volunteering is also a way in: you meet the same people every week, and it counts as Icelandic experience on a CV.