I recently spent 3 weeks in Iraq, both north and south. Some pertinent info and thoughts if you're looking to do the same.
Entry + visa - I entered Iraq through the land border with Kuwait which you, if you know about Kuwait, will not be surprised to learn was a fucking hassle. Internet says there's a public bus that runs to the border but there isn't, the only option is a taxi. I got to the border and learnt that I needed a green piece of paper from the ministry of interior authorising me to leave Kuwait to Iraq via land, and that I could only get it from the office back in the city. This wasted an entire day, as the office was closed by the time I got back to the city. I got it the next morning from the interior office next to the airport - it only took 5 minutes and didn't cost anything. Back at the border, crossing is fairly straight forward. The Iraqi side is underdeveloped and pretty dusty. Visa on arrival for all the good western passports is available at all ports of entry and cost me $77USD, and was issued surprisingly quickly. From the border town of Safwan, you'll probably take a cab to Basra. I went straight to Baghdad from the Basra taxi station, about a 6 hour drive. I imagine flying into Baghdad is far more straightforward.
Baghdad - a surprisingly nice city compared to what you may think. Much nicer than Cairo, I thought, and enough to keep you busy for up to a week, although 4 days would be fine. My favourite thing to do was strolling through Mutannabi street and the adjacent areas where the local craftsmen, book sellers, and antique dealers are. The mosques around this area are stunning and will give you a good sense of the distinct architectural style of Baghdad, i.e. intricate brickwork and blue and yellow domes. If you like modern stuff like malls, chic cafes etc then Mansour area is very nice. Museums, monuments, mosques (go to kadhimiya) and food are plentiful and will keep you busy.