There’s a lot of blogs about this, but what I can give you is a short version and a long version 🙂

Short version
Start at the Brooklyn side of the Bridge. This makes sense since you want to see the Manhattan skyline! It takes about 30 minutes to cross it but may take more depending on how fascinated you will be. For safety reasons, avoid sitting, standing, or going over the ledges! And watch out for the bicyclist on the bike lane. Lastly, check out what the weather will be like on the time frame that you plan on visiting as this area can get very hot or very cold! The ideal temperature would be around 60 deg F, without so much wind or sunlight. Always prepare for temperature challenges!
See for yourself: A comparison of the Manhattan side and the Brooklyn side
Long version
It’s one thing to read a blog and another to experience. After you visit, you will form your own opinions about the whole experience. And here are mine:
The walkway is partly made out of wood with crevices
I’m scared of heights and was surprised to have seen it. Although this information would not have prevented me from visiting the place, I would have at least prepared for it. It was not until the fourth time visiting the Bridge that I decided to cross it. Most visitors don’t cross the Bridge anyway since it will take 30 minutes away from other areas that you would want to explore.
At any rate, be careful not to drop anything!
aside from having an awesome view
The Brooklyn side of the Bridge has all sorts of fun areas such as Jane’s Carousel, Brooklyn Bridge Park, Time Out Market, Grimaldi’s, Brooklyn Heights and Promenade, etc. You can practically spend at least half of your day in this neighborhood!
getting to the BRooklyn side of the Brooklyn bridge
Using your navigation app, you can access the Bridge using the ramp by typing in “Brooklyn Bridge Walkway Starting Point” entering close to Tillary St and the Brooklyn Bridge Bistro.
Alternatively, you can also access the Bridge using the staircase by typing in “Escalera Puente Brooklyn” which is located underneath the bridge in between two walls (you will see a sign and/or the sunlight beaming through the staircase).
start in DUMBO
With friends and family, I typically would start to bring them in the DUMBO area so we can all eat, enjoy the view, rest our feet, and get comfortable with the temperature. Only after would I bring them to the Brooklyn Bridge. It’s also easier to navigate the place when DUMBO is your starting point in comparison to elsewhere.
The view in Pebble Beach and Jane’s Carousel can already be so satisfying to some that there may not be a need to go on the Bridge itself.
bikelane and ledge-safety
While it can be so exciting to take lots of photos and videos of this place, please be mindful that the walkway is shared with cyclists. Pay attention to your surroundings to avoid collisions.
And while it is tempting to take never-before-seen photos of oneself being on the Bridge, it is NOT worth risking your safety and the safety of others. Keep your feet on the ground and avoid going on or over the ledges!
always be weather-ready
I felt sick after my very first visit that I ended up sleeping throughout the afternoon and having Uber Eats deliver ramen for dinner to our Airbnb. I underestimated the temperature which was 60 deg F but then felt like 30 deg F due to the wind. This is how I learned to visit DUMBO first; to get a sense of the temperature and personal comfort.
And during one of the summers as we visited the Bridge, I brought a wide brim hat just in case someone needed it and it sure came in handy! The summer heat got leveled out by the wind, but the sun was so glaring that it got so uncomfortable!
Bottomline is to be aware of your needs and your companions’ needs to make for a great Brooklyn Bridge experience!
Happy New York City-ing!