Ahhh, I remember these days! I lurk on the 0-3 month board just to take a trip down memory lane!
In all seriousness though, my LO refused to sleep in the bassinet for the first few weeks so he slept on me a LOT. If I wanted some rest myself, I would swaddle him and lay him on a Boppy Newborn Lounger (like the nursing pillow but no hole in the middle) on the end of our bed and try to sleep for an hour or so until he woke up again. At about 2 months old, I started putting him on his Newborn Lounger then carrying it into his room and putting it in his crib with him asleep on it. But yes, he was awake in the middle of the night while my husband slept soundly and I just had to deal with it.
You kind of just do whatever you have to do at that to get them to sleep, no matter how crazy it seems! Also, try not to think about "schedules" right now because you will drive yourself crazy. It will come, but it takes time. I did the same thing so I can relate, but I was a basket case that was exhausted.
Try swaddling if you aren't already, white noise, dark room and the 5 S's. If you haven't yet, watch the DVD or read "Happiest Baby on the Block". It works for newborns!