Killing the Giant Mutant
When you go up the steps, you should be able to see a nice view of the ruined city. While this is nice, you're probably going to be more interested in the rocket launcher and rocket dispenser. Grab the rocket launcher to finally trigger the boss fight with that giant mutant that's been wandering the city providing ample foreshadowing.
This is a fairly simple fight, in concept.
You'll want to get a good shot into the mutant's glowing chest while he rampages against the building at first, so that he'll fall back and enter his routine.
This routine is pretty simple. If you do nothing, he'll just rip chunks of concrete off of the building and toss them at you. His laser eye will mark where they're going, so you have ample time to run out of the way. I suggest that you stick to the upper platform and stick to strafing back and forth.
When he throws the chunk at you, he'll expose a glowing blue spot on his stomach. You need to hit this with a rocket. Note that it's actually a pretty small window, and if you're too slow in aiming at it you'll easily miss the moment. You may want to watch him a few times and try to fire a second or so before it appears. Firing when he rears up to throw the slab at you is usually a good tactic at first though.
After this hit, he'll be wounded and distracted for a moment. He'll then rush the building and slam the upper platform with both hands. You need to be off of it when he does this. After that, his laser will search for you and pick a target. Your goal is to let him lock onto you, and then strafe away from the attack. After a few hits, he'll just grab the building and start shaking it. When this happens, he'll expose his big glowing brain. Fire a rocket into this to take off a third of his health. He'll then fall back and start throwing chunks of the building at you again.
You just need to rinse and repeat, although he appears to get faster as he takes damage, so adjust your timing appropriately. You'll really need to fire the third shot early. If you need more rockets, note that the dispenser below will produce new clips of four as you run out. If you get below four and a little time has passed, then just run down and check to see if there are any new ones waiting for you.
Also remember to reload. The rocket launcher holds 4 rockets at a time, so don't lose track of how many you have that are actually in the chamber. It's not fun to miss an easy shot because you need to reload.
Once you hit him a third time, he'll turn and die, which will expose a side entrance to the hospital, over to the left. Walk inside and switch back to your shotgun. There are still normal mutants to face.