Head a little north of Bear Mtn and just past West Point you'll find a section of rt218 called Storm King Hwy. It's only about 5 miles long but it has a nice lookout onto the Hudson and it's even more fun on two wheels. You could make a nice day of driving by linking parts of northern NJ into Harriman State Park, up around Bear Mtn, up to 218, grab lunch on the water at Billy Joe's Ribworks in Newburgh or find something across the river in Cold Spring, then down the Palisades parkway, over the GWB and back towards the city. The cruise route that was taken in 2016 for "Dub Run" would be a good place to start.
2016 Dub Run cruise route: Start at Kinnelon Bagels in the Meadtown shopping center, head northwest on Patterson Hamburg Turnpike, turn right (north) onto Clinton Rd, turn right (east) onto Warwick Tpke, turn left (north) onto Lakeside Rd, continue straight onto Jersey Ave (210), turn left (northeast) onto Windermere Ave (still 210), turn right (southeast) onto Minturn Bridge Rd (New York State Rt 17A), Continue straight onto Kanawauke Rd (106), take the 3rd exit on the roundabout onto Seven Lakes Dr (north), Continue up this until the next roundabout, take the second exit (north) onto the Palisades Pkwy (north) very briefly, almost immediately exit back onto Seven Lakes Drive toward Perkins Memorial Dr/Bear Mtn (the sign will indicate Perkins Memorial Dr), the the hairpin left turn up to the end of Perkins Memorial Dr at the top of Bear Mtn.
After this, you can choose to head on up to 218 or shoot back down the Palisades Pkwy. If you're headed down the Palisades, I recommend taking exit 2 to the Palisades Interstate Park Commission (also the Parkway Police HQ). It'll look like you're going into their parking lot out front but continue straight (south) on Henry Hudson Dr. You'll find two separate boat basins down there at different points and the last place to stop and look around is the Ross Dock Picnic Area. Stop here, take some pics of the GWB and you can even get all the way under the bridge. Just a word of caution on the Palisades Pkwy, watch out for cops on the Jersey side, speed limit is 50mph, but 75mph you should be safe passing by a cop, 80 is touchy (I've gotten ticketed for that before but also passed by many without a problem), 90 is going to be a bad day for you... (also got that but managed to get it almost cleared with a bit of help). You don't have to worry so much on the NY side, it's much rarer that I see the NY state troopers patrolling as much as I see the NJ parkway police.
Anyway, obviously this is all just a suggestion, but if you're looking for a good full scenic day of driving outside of the city, this is definitely it. There's more stuff up north in the Catskills/Adirondacks I know but thats probably too far out and I've gone on here long enough. If you are interested in a weekend up in the Adirondacks, take a look at these routes here:
https://www.gotourny.com/find-your-ride