My advice for your inaugural visit to the Brooklyn Bridge is to go early. Head out of High Street station and walk through the park towards the Brooklyn Bridge entrance. It'll be a little dark under the overpass, but it's hard to miss, especially with all of the signage and tourists heading in the same direction.
Initially, walking towards the bridge didn't feel very special. You're very much in the middle of an expressway and most likely, surrounded by a a ton of people with cameras and tour guide books, most of whom are oblivious to which lane they should be walking in. There are two- one for bikes and one for people, not one for one direction and one for the other direction. If I were on a bicycle I would have run into half a dozen people in the wrong lane. Fortunately those who were on bikes had a lot more patience than I might have had.
The closer you get to the first arch, the more special the visit starts to feel. You're not as in between cars whizzing by on either side, as the traffic is now below you, and the cables going in what feels like a thousand different directions seems to almost mask out everyone else. It also helps to look up admiring the cables creating lines and angles in every which way.
As I stood under the arch, reading the plaque, I patted myself on the back for remembering that it was Roebling who designed the bridge. I knew this because he also designed the (John A. Roebling) Suspension bridge linking Northern Kentucky to Cincinnati nearly thirty years before The Brooklyn Bridge was built. In fact, John A.'s patent of wrought iron chain links are nearly identical to the ones his son, Washington would later use to secure The Brooklyn Bridge. Cool, right? Or maybe only cool to those of us from the Northern Kentucky/Cincinnati area.
I think the view is better walking from Brooklyn towards Manhattan. You're always walking towards the impressive downtown skyline. And if you walk all the way across the East River, you'll find yourself in Chinatown, where you could easily spend the entire day sampling pork buns and picking up cheap t-shirts, sunglasses, perfume, handbags, etc. etc. OR you could turn around in the middle, like we did for some famous pizza. Don't miss out on the "Welcome to Brooklyn" sign on your way off of the bridge walkway. Because I'm new to both Brooklyn- and having a home again- it gave me a wonderful warm and fuzzy feeling knowing I can humbly say Brooklyn is my home!