Petra is defintely worth a visit.
Flying into Amman and renting a car works just fine. It's a straight shot down a multilane highway most of the way there ... well paved roads, with major signs in English. It's a bit long, though, maybe 4 hours?
There's also an airport in Aqaba (AQJ) and one nearby in Eliat, Israel (ETH). These are closer than Amman. Border crossing shouldn't be a big deal for a US or UK passport holder (not sure which you are)...I did it a couple years ago.
You can't take a rental car across, though I rented from Monte Carlo and they drove the vehicle to the border (or port in my case), and met me at the border leaving. They were prompt and professional, with reasonable rates.
Don't fly to TLV if Petra is your main goal. It's perfectly doable, but the combination of crossing the border AND being 5-6 hours away by car makes it a less-than-attractive option.
I hope this is helpful. Enjoy Petra, but also spend some time in less touristed areas, especially Wadi Rum...it's gorgeous!