Okay, so you'll ideally want a gun but a melee weapon for when you run out of ammo or your gun is damaged or whatever.
I tire of people arguing over which assault rifle is better. So I'm not even going to go there. Just pick something that has plentiful ammo and is easily maintained.
As for the melee weapon, it must be something that can be wielded in confined spaces, won't get stuck in a zombie, and won't break. Preferably usable as a tool as well. I've heard people suggest the crowbar for the tool reason. You'd strike with the flat end rather than the hook end. And it's short, and can be used as a bludgeoning or piercing weapon. Light enough that you won't get super tired and heavy enough to cause some damage. I like the crowbar idea. Remember that you're mainly using the melee weapon to buy you time to escape.
As for competing with other survivors, I think the best strategy would be to go where few are likely to go. So don't go to big box stores like Costco and such. Especially since zombies will invariably infiltrate through the many entrances, there are lots of places for zombies to hide, and in the confusion infected survivors will slip through. A better bet would be a distribution warehouse. They're usually more secure, since the public never sees them, and so there would be a few (probably metal) exterior doors and a few roll-up metal shipping doors.
The warehouse would likely have some workers in it who transformed into zombies, right? But at least it wouldn't be as bad as the mall. Again, no public, so all you're dealing with are the handful of worker zombies.
At this point you'd want to go out to gather select survivors. Don't go for the group with two people who have kitchen knives and six old ladies. Preferably you'd take people with you who are already surviving and look like their chances are good. Mentally stable is important. But with a secure warehouse with the entrances blocked by shipping containers, you don't need an army to keep watch. This was the benefit of medieval castle design - few can defend against many.
Now if the military is going to come in and kill everything, you might not want to advertise your presence. But if you want rescuers to know where you are, you could paint the roof of the warehouse where people on the ground won't see it. That way you don't get uninvited human guests but you can still get a helicopter rescue. I'd probably want to sit tight for a bit and see what happens before putting out any welcome signs. After all, we don't know just how smart these zombies are.
Definitely canned food and bottled water.
Once you have a small group, preferably friends who you already know, begin picking off the zombies around your warehouse. You don't want to worry about being mobbed every time you open a door. You should have a second-story entrance with a rope ladder or something, and whoever keeps watch does it there. This way you can look out and see if the zombies are massing nearby, and people above can help pull you up faster by the ropes if you've got zombies clawing at your feet. Like someone said earlier, zombies won't reproduce, so however many zombies there are now are the numbers you'll deal with in total.
You're probably in a kind of industrial area because of the warehouse site. So you have more tools, materials, and industrial vehicles around while you have less food, clothes, gas stations, and shops. But you also have fewer people around.
A side benefit is that unlike typically open commercial and residential areas, industrial zones tend to be compartmentalized by sturdy fences and walls. What discourages a burglar is probably pretty good at keeping uninterested zombies from wandering around. This means if you go next door to a plastics manufacturer, the parking lot probably has a fence around it and so you won't get zombies coming in from all sides. And the joint is probably just as secure as your warehouse (a couple strong doors and a shipping door or two, no large floor-level windows). If you wanted to, you could clear out the areas immediately around you and lock them up, so you know the only zombies around are the ones in the street.
Then again, that's a lot of work and risk for a little security. You'd probably be just fine living in the warehouse for a while.
By the way, gasoline from a gas tank wouldn't be good anymore after a couple months. Gas stations have more sophisticated storage systems, and their gas could be good for years. But unless you get the oil refineries going again your car will be useless in a couple years. Electrics are viable if you have a house-roof sized solar array and two cars. Charge one while the other is in use, and in emergencies you have two. Actually, it would probably be better to have a few sitting around charged up
But in the long run, if people aren't making bullets and cars and gasoline and solar panels, we're pretty much screwed. So it's important that the zombie threat be taken care of within a year or two, otherwise our ability to fight it would be so diminished that we might never win.