This is a nonsequitur if I ever saw one.

It doesn't matter whether I plan or not, because you'll still need to plan either way.

For me, planning is a matter of circumstances. On local rides I generally have a sense of what I want to do, and don't care if I get lost -- or to put it nicely, go exploring. It's all riding so it doesn't really matter where I do it, or even if I know. OTOH on longer rides with a specific destination, and riding range or time to consider, I do some basic route planning and at a minimum, have some sense of where I'm going, how I'm getting there, and how long it's take.

When doing multi-day touring, I'll roughly plan day to day, but I carry a map (you know, folded paper) and feel free to take detours if something interesting happens along the way.

To tho OP, you're starting in the Bronx and there are a number of good routes that take you out to City Island or up to lower Westchester. If coming to Westchester you can roughly parallel one of the 3 Metro North lines, and if you get lost or tired, work your way to the nearest station and Rail back to the nearest stop to your home. That opens up more nicer riding, and should relieve worries about getting lost.

Metro North is very liberal about taking bikes on off peak trains, but you'll need to go to Grand Central Station and buy the blue permit (lifetime, and cheap) because some conductors will ask for it.

BTW- I've ridden as far as Montreal, Philadelphia, and Rochester, riding no map, and just kind of winging it and keeping to a general direction.