Baby carriers come in many more forms than just Baby Bjorns, although you will see those everywhere. Many baby carriers are easy to sew, even for beginners. I have found mei tais, which are good from birth to toddlerhood, to be very easy to sew. Look "mei tei sewing tutorial" up on Google. Ring slings, podaegis, and wraps are also easy to make. You do need special supplies, like rings, for certain types of baby carrier though.
If you are planning to give cloth diapers a try, sewing them is fun and not at all difficult. Like with baby carriers, there is an abundance of internet tutorials. You'll need snaps or velcro, elastic, and speciality materials like PUL (the water-resistant layer) and flanel or bamboo.
Bedding for babies can be rather boring. You can make your own with cute fabrics, or just buy plain white cotton bedding and embellish it with appliques. You can also tie-dye your baby's bedding, which is fun and easy.
Beginning sewers may like to tackle simple baby pants and kimono tops. Advanced crafty moms can sew anything! Just keep in mind that newborn size may not be big enough even for tiny babies. You may like to make clothes in slightly bigger sizes. Baby gowns are another popular item that moms who sew make for their babies.
I've seen the cutest baby bootie patterns around the internet. These are easy to sew and are perfect for any scrap materials you may have lying around. In particular, the soft flannels you may have used for diapers are great candidates.