J.,
Congratulations for your baby boy! I teach parenting classes and do a lot of a child advocacy. For a few years before my oldest daughter was born, I provided some in-home counseling for families and worked a lot with helping parents sleep train their infants. People have often commented about how fortunate I am to have children who go to bed so easily, but a lot of work has been invested in getting them to that point! Given this, I hope you find my opinion helpful.
Infants are not wired to "put themselves to sleep" until 4 months of age. Some newborns will lay down and go to sleep on their own depending on their surroundings, but that depends more on temperament than ability. They live in such a noisy, bouncy place while they are in utero that the quiet and stillness of a bassinet or crib is unsettling. They have been bounced to sleep for many months. Is he gassy? Some gassy babies don't sleep well without body warmth or movement because these two things relieve gas pains. When he sleeps on you, he hears all of those familiar sounds; your heartbeat, gut sounds, etc. and his stomach is warm against your body.
I, too, have a newborn son (4 weeks) and we have to hold, rock, or swing him to get him to sleep. After I nurse him during the night, I will hold him to burp and he falls back to sleep with me patting him. He will, also, often wake up when I lay him in his bassinet. I plan on doing the same with him as I did with his older sisters - work on him falling to sleep on his own after 4 months of age.
I don't know what your circumstances are but some people stress about what they are "suppose" to do. They listen to people say that infants should sleep in a crib and go to sleep on their own. In reality, the parents should do what is best for the child and the parent provided it is safe. What is wrong with an infant sleeping in a swing or a bouncy seat until he/she can no longer safely fit in one? Some babies must sleep in their car seats, bouncy seats, swings, etc. because of acid reflux or other medical conditions. Provided it doesn't pose a risk to the baby, who cares where/how he sleeps if he is getting sleep and you feel like a good, rested mother?
I think it really depends on why you want him to fall asleep on his own. If you need him to because of your lifestyle then I would try to be patient until he reaches 4 months. However, if you just want to avoid haivng to rock him until he is older, like your daughter, then I would relax and not stress. The great thing about sleep training is that you can do it at 4 months or 2 years and it works within a week. Some people, like me, enjoy rocking their babies. Once my girls were about 5 months old, I would nurse them, rock them until they were sleepy, then lay them in their cribs to fall asleep. The only exception to this should be when they are sick or traumatized. As adults, we would like to be soothed to sleep when we are not feeling well!
If you enjoy rocking, holding, or whatever you do to get him to sleep, I would continue to enjoy it and not worry about his sleep patterns until he is a little older. For now, give him ample time everyday to lay down by himself to look around and stretch out. I you don't enjoy it and really want him to fall asleep on his own, you could try to give him a pacifier (if he takes one) and pat him on his side or bottom while he is laying down until he is old enough to go to sleep on his own. Please just know that his inablity to fall asleep on his own right now is not on behalf of anything you are doing - it is natural and he needs to get a few months older.
I hope this was helpful! Good luck and enjoy that precious baby!