I saw in a comment you're coming from downtown. The most straight forward way is to bike up Newark Ave and then turn right on Palisade. Newark Ave has a protected bike lane, so it's a safer ride, although you have to cycle up the hill. Palisade does not have a protected bike lane. It doesn't even have a separate bike lane. The "bike route" on Palisade is just sharing the one-lane road with cars. There are signs that say "Bike can use entire lane", or something like that, but the law is one thing, and people are something else.

Personally if I cycle from downtown up to heights, I prefer to turn right on Central Ave, not on Palisade. Palisade had too much road work done in the past and the road hasn't been repaved. I find Central Ave to be a smoother and safer ride (if you do choose to turn right on Central, ride on the sidewalk for the one block stretch from Newark to 139). Central doesn't have a separate bike lane, so you'll still have to share the one-lane road with cars. Personally, if there's traffic, I often ride slowly on the sidewalk.

Most roads in the Heights don't have a separate bike lane. None have a protected bike lane, as far as I know. Feel free to explore around the Heights. You can also cross into Union City if you want. If you have any of those cycling apps (Strava, Ride with GPS), I think you can see popular cycling routes that people take, so you can use those as a guide for where to cycle.

If you have difficulty cycling up the hill on Newark Ave, you could always use take the PATH from Grove to Journal Square. Both stations have elevators for your bike. If you do take the PATH, it can be a bit tricky to figure out how to enter/exit the stations with a bike, but once you know where to go, it's an easy option. And from JSQ cycle on Central into the Heights. Feel free to ask all of us if you have more questions. You should also follow BikeJC on instagram/whatever socials they have. I'm not involved with them, but they do great work.