I checked out the Skyvector.com site and its pretty darn cool. It's just a little overwhelming. So much information on a chart and I can only understand half of it.
If you drag the view over to the left on one of those skyvector sharts, you will see a legend that defines all the markings, etc.
But you may want to start with a sectional that is not quite as cluttered as the New York chart!

I'd like to try out flying from Stewart Airport (SWF) and finding a local landmark around here like the Tappan Zee Bridge and then landing at Westchester (HPN)
In case anybody wants to see what I'm talking about...the Tappan Zee bridge is the Hudson River crossing between Nyack and Tarrytown (GPS 41.07/-73.882). It should be easy to find once you locate SWF. Just go south and then slightly east until you see the Hudson River.
What would be the proper way to plan this flight? Would i still head to the Carmel VOR and then estimate the heading out to the Tappan Zee Bridge? (Looks like it would be about 240), and then just guess the heading again to HPN?
You can do it any way you want, but having flown (RL) in that neck of the woods for years without ever using GPS and rarely using VORs, I think you're making a lot of work for yourself for such a short flight. That should be a straight shot, and there are so many vidsual landmarks it's a no-brainer.
Without a chart, you could just use Flight Planner to plot a direct course and use that heading. Note some landmark at or near the river, and make sure you hit that mark as you cross the river. If you miss it, that means you're drifting and need to change heading slightly to compensate.
Tip: the Bear Mountain Bridge, north of the TZ, would actually be a better landmark...it's about where you would cross the river flying direct from SWF to HPN.
Mind the restricted airspace over West Point, though.
I've never actually landed at HPN in RL, but between the river itself and the big lake NW of HPN, it's darn easy to find when flying direct from SWF.
I would get the ATIS and call HPN tower once I was passing Croton Point, which is that spit of land on the east bank, north of the TZ Bridge.
the TZ Bridge or the city of Tarrytown might make a better reporting point in RL, but you'll end up going out of your way to use those.