Do I actually have to tell someone in the group, "this is my home group now?". Or do I just know it for myself? Also, should the sponsor I ask attend that group or doesn't that matter? If the group only meets once a week that is ok, correct?
Your home group is where you will do a lot of your service work (making coffee, setting up chairs, greeting, chairing, etc) and where you will attend all the meetings - so if they meet twice a week you will do your very best to be there. It's a place that, once get settled and get to know others, you will be missed if you're not there. By taking on commitments, it helps to ensure that you show up.
As for telling someone in the group, yes, you have to tell the secretary of the group - just ask someone in the group who the secretary is. They will take down your sobriety date and a phone number usually. That's it.
As for the sponsor - it can certainly be from you home group, but it's not gospel. I would say of all the people I know, most of their sponsors are in different home groups. In a sponsor you want someone who has worked the steps, is active in the program and has a sponsor themselves (my opinion at least). You're not looking for a best buddy - you want someone who has what you want, as they say.
Most groups have two meetings a week, or do a double header on one evening. At least in this part of the world. If your home group meets only once a week, that's fine too. Just hit other meetings as well. We all do.
Find a group that you feel comfortable in - some people like larger groups, some like smaller. My personal opinion is to find a group that is literature based (they work out of the big book - not just open topic discussion meetings where you talk about whatever). If you're in a big city or town, you will have lots of options, if you're in a small town, you may not have many.
Regardless, it's important to have a home group. You need a place to be centered in, where you can do service work, where you get to really know others, and where you can settle in when life throws you curve balls, where people know you.
Good luck!