I went to Mongolia in 2005 and spent about 18 days on the ground there. We were in UB for just about 3 days total, and spent the remainder of the time driving around the countryside in a giant loop (about 2200 km altogether). We first went south to the Gobi, then west, then north to wind up at Lake Khovsgol.

If you have only six days then it will be tough to do much (IMO the farther you get from UB, the better). I do NOT recommend tight scheduling of domestic flights. The one domestic flight we did, from Mörön back to UB, was delayed about 15 hours; I missed my flight out of Mongolia and had to pull every trick in the book to get out of the country and back home ($2600 poorer than I should have been, and in economy the whole way vs. the F award seats I had booked ). The airline we were on, Aero Mongolia, had exactly one plane, and they apparently needed to take some local honcho's parents somewhere at the same time our flight was scheduled. So they just left all of the 50+ pax sitting there all night while they did it. This is the kind of place Mongolia is.

Getting a jeep/van with a skilled and experienced driver is a must. I did this online with a local hostel owner/tour agency before I arrived. You can do it on arrival if you want, but it can take a couple of days to find a guy and get outfitted for the trip, so you may not want to waste time doing that if you only have six days.

I set up my trip via Idre's Guesthouse. They did a good job setting up the tour, but the hostel itself in UB was pretty dreadful (although for $5 a night I had a hard time complaining). For our countryside trek in a Soviet 4WD van, we had a driver (an absolute wizard), translator/guide (quite poor), and a cook (good). The whole thing was an experience, for sure.