Reinforcing their defenses and economy
The biggest threats to any society are their bellies and their blood. If they can't maintain good foodstocks in the long run they'll die, and if they can't protect themselves they'll die. As such, they need to reinforce their home base.
Most government types will be moderately functional if the society is small and food and water is common. People can negotiate out any problems directly with leaders and the low distance means it's fine.
Nuclear bombs, radiation, and waste are big early threats.
If they are hit directly by a nuclear bomb they are dead, but if they get the edge of one, or the waste, they'll die. In the short run they should be reinforcing their base. It's unlikely that anyone will be aiming directly at them, and post apocalypse it's unlikely anyone will have the infrastructure or knowledge to maintain or shoot a nuke, but it's very possible in the next couple of weeks someone will fire something.
Radiation also falls off rapidly.
A nuclear explosion produces a complex mix of more than 300 different isotopes of dozens of elements, with half-lifes from fractions of a second to millions of years. The total radioactivity of the fission products is extremely large at first, but it falls off at a fairly rapid rate as a result of radioactive decay. Seven hours after a nuclear explosion, residual radioactivity will have decreased to about 10 percent of its amount at 1 hour, and after another 48 hours it will have decreased to 1 percent. (The rule of thumb is that for every sevenfold increase in time after the explosion, the radiation dose rate decreases by a factor of 10.)
As such, they should be reinforcing buildings to act as shelters, securing air gaps, and generally making sure their core base is safe. If they can hide inside from radiation for just 48 hours it becomes a much smaller threat. There's still a cancer risk, but nothing like what it was like right after the nuke.
In the middle timeframe, humans are the bigger threats
Soon, survivors who are starting to run out of good supplies are gonna look at your base and think "We deserve that." Your walls and reinforcements along with good guards should help deter them. You want your base to be fairly solid so they can't just blast you out, and you want the surrounding area to be well scouted so you can smash any threats.
This is also the time when you do as much scavenging as possible, and reinforce your supplies. Having a wall to do so is essential as you need somewhere safe to run to when things get hot.
In the long run they need to start generating supplies
Over 5-10 years supplies will start going bad. Some will last longer, but to remain functional as a community they need to start producing food and machine parts and other useful supplies.
This means their city needs to be stable. They need walls and guards to prevent someone from stealing their supplies or stocks. Walls and guards remain the most important investinvestment of time.