We went to Amman in August and, tbh, it may be cheaper to hire two street taxis rather than a minibus. We used both and the minibus was far more expensive: a trip to Petra, including entrance, cost $155USD each (booked as a group of twenty through a Lebanese company so unfortunately I don't have the contact details - friend of a friend situation). A trip to Jerash in a taxi was 30JD total (with Khaled, see below).
The only advantage I can think of in using a minibus/tour company to get to Petra is that they may get a discounted entrance fee (like we did) otherwise it's really expensive (I think 90JD for one day if you don't stay overnight).
I would honestly get two taxis and bargain with the drivers for all other places. Agree the price before you get in. I also recommend (if you are fit) going to Wadi al Mujib, which is next to the Dead Sea (about 1hr30-ish) from Amman and is amazing. It involves a hike through a river and even women should wear shorts (we did and got funny looks, no comments, in Amman as we left but no comments or looks at the Wadi - they knew it was necessary to do the climbing) and all should wear really sturdy shoes. It's 16JD to get in and so worth it.
Our taxi driver was great, with excellent English. His name was Khaled and my friend has his business card with telephone number. Unfortunately she's not in the UK at the minute but I can get you the number within a week or so if you would like? He also recommended a good restaurant, ate with us and chatted about the country and his lifestyle, which was really interesting.
In Amman the train station is also wonderful. It's officially still open but no trains run so it's gently rotting. There's a caretaker and you can just wander in and look around at the sheds (our driver got him to open them for us so we could see the workshops - tools just left as though they were coming back the next day) and the Royal carriage etc. It's like an old film set - everything dates from the 1950s with modern air con strapped to the sides.
Sorry, this is quite long - I just had a really great time :)