My son slept in a side-car co-sleeper until 14 weeks when things went downhill. I tried putting him in his crib for the first sleep-stretch of the night, and that did the trick (instantly started sleeping 8 hour stretches). When he woke, I would bring him into our room, nurse him, and then put him in the co-sleeper for the rest of the night. Also, he never slept on his back, only his belly. When I decided to let him sleep on his belly full time, starting at about 7 weeks, his sleep got infinitely better.
My daughter slept in a regular bassinet until about 10 weeks. I think she felt too confined by it and the mattress is thin, like other moms have said. When I switched her to her crib, she too began to sleep much better. I had to elevate her mattress though, because she needed the incline (put a pillow under one end). She sleeps more like your son, even still does not sleep long stretches (almost 9 months), so I just do whatever it takes - if putting him in the stroller next to your bed means everyone is sleeping, I wouldn't worry about it. You could also try using a pack n play and beefing up the mattress by putting a baby quilt on top of it or a fleece blanket (my daughter LOVES fleece and falls asleep instantly). Once he's rolling around a lot (6 months), you can try him in the crib, and by then he'll be old enough to remove a night feeding or do a little sleep training. Don't worry, babies start to sleep much better by 5-6 months, and then many of them are sleeping 12 hour stretches by a year.
Have you read Healthy Sleep Habits, Happy Child by Weissbluth? This book helped me tremendously and can work for all sleep-related parenting philosophies. Start learning about sleep now so when the schedules emerge, you can identify them right away. Good luck!!