I commuted from downtown Jersey City to Staten Island by car for several years — don’t be scared off, it’s really not bad. The Bayonne Bridge is literally always empty. Your main issues will be the Staten Island Expressway, if you must take it, or other roads on SI being jammed at rush hour, depending on where you’re going. An occasional backup on 78 in Jersey City, but you can take Caven Point Road to get around it (and save a few bucks on tolls).

I second the recommendation of Bergen-Lafayette. You can find street parking, and there’s the turnpike extension on-ramp to Staten Island right over by the Liberty Science Center. I live in the neighborhood and drive to SI all the time to see family (and sometimes for work obligations). It’s usually a breeze.

Downtown, big buildings you’ll have to pay for parking. If you rent in a small building in an area like Paulus Hook, you can park on the street, though sometimes you’ll have to walk a few blocks. That’s where I lived when I commuted to SI. It’s doable, and hipper than Bergen-Lafayette, but Bergen Lafayette’s a little easier/cheaper.