I've been to two different kinds of groups and they were both very different. The hospital I delivered at has a weekly group, I only went once early on and although it was great if you were having issues, it wasn't exactly what I needed. It was led by a lactation consultant and basically you weighed your baby when you got there then nursed and weighed your baby again to see how much they gained after a feeding. She would walk around the room and answer questions if anyone had any. It probably was a great group if you were struggling with a latch or weight gain ect, but it wasn't what I needed so I never went back.
Then I found a local La Leche League group and it was exactly what I was looking for. It's just a group of breastfeeding mothers that come together to share information, talk about anything and all things breastfeeding related, ask questions, and basically just support each other. Some local chapters have formal topics at each meeting even though of course all questions and concerns are addressed while others are much less formal. Sometimes no one had any real questions so it was kinda like a baby playgroup/new mom social group. It was a great way to meet other like minded moms and I went religiously for a very long time mostly for the social aspect even when I really didn't have any questions.
Other groups might operate similar to the LLL meetings and not necessarily be like the hospital one I described. Breastfeeding support groups are a great way to meet other new moms in the area and I learned so much about breastfeeding from going to them. Different things come up along the way and it was great to have a place to get information that was trustworthy.