Universally most babies don't like to be put down for long. It's genetically to the advantage of the human species (evolution-wise), since babies who are put down in unsafe places are sometimes eaten by wild animals! That's why they fuss if the range of "normal" behavior for all babies, it is COMPLETELY normal for a one to three month old to be either asleep, eating, or crying if not held.
Your son needs to sleep with you at night, he is probably pretty used to falling asleep next to another warm body. So as far as his napping goes, you can either let him fall asleep in the baby carrier, or you can help him start learning how to sleep on his own. Try swaddling him, to mimic the feeling of being held, and then putting him down. Stay with him and rock him, sing, or stroke his face or hand until he settles down. Babies this young simply don't have the ability to calm themselves yet, so it's important not to let him "cry it out."
During the second half of their first year, babies begin a developmental process I call "hatching," in which they start to realize there's a whole world out there apart from mom. Baby's understanding of himself evolves from "Mommy and I are one" to "I'm different from Mommy" to "I'm an individual." Besides the intellectual desire to be "me," he begins to develop the motor and language skills that allow him to do things on his own.
But in order to achieve healthy independence, baby must first feel secure that his parents are there to care for him. The baby who misses the close connection of "Mommy and I are one" likely will find it harder to move into the "me" stage, and if pushed to give up his natural, healthy dependence on his parents too early, he could easily become a clingy, insecure baby.
One of the myths about separation anxiety is that it's caused by mothers who are too attached to their babies. This is a carryover from the 1920s-era spoiling theory, which maintained that holding a baby a lot, feeding on cue and responding to her cries would create a clingy, dependent babies. But current research shows that strong attachment fosters independence, not dependence. Babies who are the most connected early on have such strong trust in their parents that calm reassurance from their parents gives them the comfort to try new things or explore unfamiliar territory, knowing that help will be there if they need it.
But what can you do to get the breaks you need? Some parents find that a baby carrier or sling is a good compromise. They allow you to carry baby with you while you get things done around the house. My daughter was a lot like your son, and I remember marveling at the fact that even as I bobbed up and down while doing the housework she never made a peep in the baby sling. This is also the time to call some of those people who offered to help with the baby—friends, family or neighbors to come over for an hour or two and hold your son while you shower, return e-mails, run errands, or just take a well earned nap.
It's also important that you establish a routine early on. And that you or your spouse bathe and massage and clean and feed your baby and not let a maid do any of these. Babies need to look into their mothers eyes and see her face when they are being bathed massaged and fed. This increases their security and strengthens the bonding process. The first few months are crucial for bonding and is when your baby to fall in love with you and you with your baby,
It will take time for your son to learn to fall asleep on his own and to be content without you; it's a skill he will spend most of his first 6 months (to a year) mastering. So be patient, seek out help when you need it and, even as you fantasize about a whole 30 minutes alone, remember that these early days and months fly by very quickly. Before you know it, your son will learn to crawl and walk, and you'll be running to catch up with him!