I have one. Flew some friends to Pittsburgh for a bris (2.75 hours each way), proposed to my wife on the plane, went with my wife for thanksgiving weekend to Newport, RI, and flew into JFK not to mention the umpteen trips up and down the Hudson River I've done.
You can fly cross-country, but in the basic Pipers and Cessnas, total flight time would be about 40 hrs or more (RT). The most these planes can fly is about 5 hours depending on how much you load up on fuel and a myriad of other factors. So, you're talking minimum 8 stops along the way. It would be a great "road trip" type deal. Other factors to consider include weather and cost. Assuming you have luggage and full fuel, you really can only take 2 people at a time and that would set you back about 4 grand just in flying costs even if you get what I consider to be an amazing rate of about $100/hour. So this would not be to try and save money. Flying is an expensive hobby but not too expensive if done properly. All of the metrics I have mentioned change considerably depending on what type of plane you fly, whether you own, rent or join a flying club (more on that later), how often you fly and what types of licenses you get.
If you are looking to get your private license which entitles you to fly a single engine piston aircraft during the day and at night in relatively good visibility and not for-hire, then my recommendation for 99% of people is to join a flying club. Basically, a club is like a timeshare. The club may have 1 or more planes for use of the membership based on a (mostly) first-come first-serve scheduling system. Most clubs allow you to take the plane for up to a few days with a maximum cap of total "vacation days" per year. Basically, as long as you're a mentsch about it, they will be very accommodating. Clubs usually charge an entrance fee of about $400-$600 which is non-refundable. They also require a deposit of anywhere from $500-$5000 which is refundable upon your resignation from the club. There is usually a monthly maintenance fee ranging from $50-$150 a month. This does not include any flying time. Rates on flying are either "dry", which means that you are only paying for the plane and you have to pay for fuel or "wet" which means that the rate includes fuel (fuel is anywhere from $4.50/gal - $8 or $9/gal and these planes average between 8 and 10 gal/hr at cruise). One of the great advantages of joining a club is that they charge based on Tach time (which is determined by a meter hooked up to the tachometer and at cruise will usually match actual time flown hour for hour) as opposed to Hobbs time which is completely time based - if you turn on the engine, and an hour later you turn it off, you will pay for an hour even if you just idled on the tarmac (or sat in a long taxi line at JFK) the entire time.
A few other pieces of advice: Make sure you have all the money you are going to need to complete training before your first lesson. For me, start to finish I spent $7500 but I was extremely lucky since I b"H got an amazing instructor and was able to do most of my training in one summer (30 hours worth) upstate. A more realistic estimate is at least $10k. Also, make sure you have lots of time to do the training. Try for at least once a week for a couple of hours. If you only plan on doing an hour a month, then the whole thing will take you longer and cost more money since you will have to re-learn almost everything each time you go. It will also make you feel less accomplished. I flew solo for my first time after officially 8.5 hours (but the first 1.7 hours I wouldn't count in real terms since they were intro flights and didn't teach me much) and I felt amazing. If it takes you a year to solo, you're gonna feel pretty down (BTW, 8.5 hours is only doable with a really confident instructor in a really non-congested area like upstate NY. A young instructor won't take a chance with you at such low time and even a confident instructor won't be able to get all the work that's necessary to fly in controlled airspace done in that time).
BTW, there are a bunch of frum pilots out there and we all love giving each other support.
All-in-all, flying is amazing, but don't treat it as a mode of transport unless you plan on getting a few licenses (Instrument rating, commercial rating and maybe even more) and your own plane (so that you can tailor it to your needs).
I know that was a doozie of a post. Feel free to ask me anything else