Welcome to the site.
First for the rod and reel. Anything from a light-medium 7 foot rod to a medium-heavy 9 foot rod would work. I spend most of my time fishing for bluefish and striped bass and use a 9 foot medium-heavy rod since the fish tend to be a bit bigger. If fishing piers and docks in bays, a 7 foot rod would be great for fluke, sea bass, porgys, kingfish, etc. Here is a link for a basic rod and reel combo that I'd use.
I don't have this exact rod and reel, but a similar one. It'll be fine for pretty much any fish that you'll encounter in our waters. I have a similar rod that's my go to that I've been using ever since I started fishing in 2006.
Second, I don't spend too much time down the shore so I can't really speak to piers down there. I've got a lot of experience with piers and such in Hudson County which I can share with you. Then I do an okay amount of fishing down in the Brick, Lavallette area. Can't help you with Perth Amboy area.
Third, you'll probably have to lower your expectations with the 60 pound fish, especially from shore. Very, very unlikely you'll catch anything other than a shark that's 60 pounds. A black drum or striper are possible, but I think I'd rather play the lottery than make bets on when I'll catch a 60 pound striper. The shark is possible. If you search this forum there are a couple topics about fishing for sharks from the beach. Usually done in late August or early September. Bunker or really any kind of small fish (small bluefish, kingfish, porgies) will work. I've never done it, but it's on my list. I'd imagine the 9 foot rod should be okay for that. You'll have an uphill battle reeling it in, but assuming you have some heavy braid, shouldn't be out of the question.
Hope that helps. Any other questions?