You know the most important thing you need? A flashlight. Why? Because most superhero work takes place at night since most crime takes place at night. At least most crime superheroes fight. White collar crime takes place from 9-5 but superheroes don't give a shit about insider trading usually so whatever. I think I'll go use my accounting knowledge to create a new hero: The Auditor! The Auditor uses his keen accounting skills to detect fraud, thereby saving money for ordinary citizens! I should pitch this to the AICPA; I could probably get them to fund it.
Anyway, I'd say get one of those big heavy flashlights like cops use. One of those big Maglite mothers that when you load it with batteries also works as a great club. That way if the guns and knives I told you to buy don't work, you have something else to rely on. You should also have a backup flashlight that's smaller, like a penlight for when the big honking flashlight is too cumbersome.
Another good thing to have: a cell phone. Why? Well these days there aren't pay phones anywhere so after you beat up the bad guys and tie them up, how are you going to get the cops there to pick them up? Or conversely if the bad guys beat the shit out of you and you're bleeding to death in an alley you need to be able to call for an ambulance.
Now since you don't want people to know your secret identity, get one of those little disposable prepaid cell phones and for God's sake don't buy it with your credit card! At least that's what they always do on "Burn Notice" where it seems like the superspy runs through fifteen cell phones per episode. (Incidentally, you should watch that and "MacGuyver" and stuff because you can learn interesting little tricks like turning a can of keyboard dusting spray upside down can freeze a lock so you can smash it open!)
If your disposable phone is good enough it might have some nice apps on it like say Google Maps so you can figure out where you're going. If not you can buy something to do that I'm sure.
Since I mentioned it, you need something to tie up bad guys with. Rope might work but it's not that portable. You can probably buy some of those zip tie dealies the cops use online--like this. If nothing else, get some sturdy duct tape and tape the wrists and ankles as tight as possible.
Something else you might want to look into is a lockpick kit. They probably sell those online too. At the very least keep a thin wire-type thing around in case you need to pick a lock and get in somewhere. You can find out how to do that online. I know because I looked it up once for a story. Plus Russell Crowe does it in "The Next Three Days" by looking it up online and movies never lie, right? Anyway, I'm just saying you should do that because if you're running around playing World's Greatest Detective you might need to get behind locked doors. Try not to misuse those skills though.
This is probably just scratching the surface. If you have a decent budget you can get all sorts of stuff like nightvision goggles and so forth. Maybe you people have some ideas?
And let's not forget the mos essential thing any superhero needs: Shark Repellent!