The key is to stick in smaller groups but not to exclude other people. A group of 20 people with a wide set of skills should allow for you to survive for months. The smaller the group, the easier to get overwhelmed; the larger the group is the harder it is to keep mobile or to provide for every person.
The only problem will be the initial panic. You want to build up your supplies as quick as possible, but the likelihood of you being the first person to think of going to X store for non perishable food and water is highly unlikely. Most people will have enough supplies in their house or within their community of houses to survive without needing to leave for 1-2 months, so this can be an option for people who don't live in populated areas. However this initial panic will also kill off a lot of idiots, so the resulting population is luckier and smarter, hopefully smart enough to realize that they can't survive alone.
Beyond 1-2 months assuming that civilization has collapsed, the first thing to do is to look for shelter with adequate construction, access to medical facilities, cooking supplies, internal electricity, communications, and heavy set doors. A few good examples include: Hospitals, high rise buildings, high schools, military bases, factories, etc. However all of these depend on specific situations, and may be better or worse depending on what you're looking for. Military bases may have weapons, but they're large and would limit you to using at most 3-4 buildings without spreading your defenses thin. Hospitals may have a seemingly infinite amount of medical supplies, but the chances of finding a hospital that is both empty and not in a densely populated area or zombie hotspot is a long shot. You should choose your building based on it's qualities first, then location, then size. My personal choice would be a high school, because the have medical facilities to cover any minor injuries, cooking facilities to feed around 1000 people, an emergency generator, and very thick doors.