Hello,
first of all, I used to live in Jordan for the past two years so if you have any specific questions don't hesitate to send a pm.
Two weeks is quite some time to see a lot of beautiful and not so beautiful things. Next to Tel Aviv/Jaffa and Jerusalem I do recommend you go and visit Akko, located close to Haifa (which has a nice museum of art - the museum of modern art in Tel Aviv is superb by the way! Do not miss "Bauhaus" if you are into architecture). Akko hast a wonderful downtown. Haifa is famous for its arabic neighbourhood. I always recommend to go and visit Bethlehem and Ramallah, Westbank, as well if it is possible for you. Different world.
Public transportation in Israel is easy and very reliable! In contrast to: Jordan.
The only reliable public transportation in Jordan, which can be used without a basic knowledge of arabic is - in my opinion - a company called "JETT". They do have a Bus which will bring you to Petra. You could also rent a car - otherwise it might be difficult to get around. Basic transportation within Amman is done by Taxis (very cheap, insist that your driver activates his "adat" (Meter), it should start at 0.25)
Petra is great. You could go to Wadi Dana too - it is my favourite place in all of Jordan. I can recommend a Hostel if interested. Overlooking the valley.
Are you planning on crossing into Jordan using Allenby Bridge (in Jordan it is called King Hussein Bridge) close to Jerusalem? If yes - please inform yourself about visa conditions. You can't get a tourist visa when entering Jordan at the bridge - you have to get it in Jerusalem in advance (valid for German passport - can imagine this is true for a lot of different passports). Since you are interested in Petra you might decide to use the southern border crossing (Eilat/Aqaba). There you can get a tourist visa! If you are only interested in Petra I recommend to do it this way since, as mentioned before, public transportation/transportation in general is easier in Israel.
Enjoy it. Very interesting region!