The northern Israel-Jordan border crossing is probably as easy a route as you can get for the independent traveller. The Jordan-Israel border is fraught with inconveniences. The most common crossing is of course the King Hussein Bridge which connects Amman to Jerusalem, the problem is that you don't get an exit stamp when leaving Jordan as they don't recognise the West Bank as Israel. Doing the trip vice-versa from Israel incurs a higher exit tax as you are charged by both the Israelis and the Palestinains.
The southern border is relatively straight-forward apart from the awaiting expensive taxi ride from the border to Aqaba, unless you are heading/coming to/from Egypt or visiting the expensive Red Sea destination of Eliat, there's little reason to do it. The exit tax is also a little higher there.
Here's how I did the trip and why I think it's the best way. Get yourself to Be'et She'an, unless of course you are coming from Haifa, Nazareth or Tiberias in which case you only need to be dropped off at the Jordan River Crossing junction, from then on it is an easy 4km walk downhill to the border.
Pass through the waiting taxis, in to the terminal and pay your exit tax at the exchange place (60 shekels as opposed to 90 (Eliat) or 120 (King Hussein)), try to keep 4 shekels before changing your money. You'll need it for later. Head to the immigration section with your passport and proof of paying the exit tax - the queues will likely be relatively short. Walk through duty free and wait at the bus stop, pay 4 shekels to board the ridiculously short bus journey over the bridge (they won't let you walk for some reason).
Get off the bus, go into the Jordanian customs office and change money and fill in your immigration card, then get yourself in to the queue of impatient Israelis f(there's a good chance you'll be able to knock them about a bit with your backpack - always a good line of ammunition in a free for all queue), once you have your visa in yourpassport, get into the next queue for the border officer, smile, answer a few questions and you have your stamp.
Then it's out and into the warehouse like building, put your metal and bags into the metal detecting machine just as you would the airport and be prepared to answer questions about medications and electronics (although they didn't consider my laptop enough of an object to probe, they seemed intrigued by some old American ladies stuff.)
Now you and your things are into Jordan and you will be bombarded by hundreds of taxi drivers, they claim to have some system going with a guy writing down stuff on the pad. The prices they quote are ridiculous and they don't go particularly low when you barter hard with them. Most of them are downright liars and will tell you things like you can't walk further, it's not allowed and multiply the distances of the nearest city Irbid.
They'll keep it up for the length of the road leading out of the border crossing with taxis cutting into you and beeping up to the main junction (probably about 1,5km). In reality, once you leave the border compound - the police at the top are most likely going to wave down the bus which regularly travels from villages in the area to the nearest town of Irbid.
The price is a mere 500fils and not the ridiculous 20JD that the taxi drivers want. 30 minutes or so and you are in Irbid, take a taxi across the city for about 1JD to the main bus station and hop on a bus to Amman which varies from 1 to 3 JD depending on the service.
People in Irbid are friendly and willing to help out, it's just getting past those initial gitty taxi drivers that puts most people off, overpaying for a mammoth taxi drive seems to put many a tourist in a miserable "I feel ripped off" mood whilst in Jordan and there's really no need.