I didn't start babywearing until a bit later on with my first child. At around 2-3 months I started using a Moby and Bjorn, and it wasn't until around 6 months that I started using a ring sling and purchased my first woven wrap. So I was really excited to start babywearing from the beginning with baby #2 (plus, I knew I'd kind of need to with a 3.5 year old to keep up with, too!). Here's a little primer on my experience and recommendations for babywearing from the start.
If you plan to babywear in the hospital, know that you will only be able to do so in your room in most cases. Most hospitals require babies to be rolled in the bassinet if they are out in the hallway. Ring slings are my favorite carrier in general, and are great for newborns. I decided to pack a ring sling in my hospital back to use there. Ring slings are pretty quick and easy (practice with a baby doll at home first if you are not well-versed, and read our post with tips and tricks!), small to pack, and don't require tying around your midsection. Woven wraps are also a good choice if you are having a vaginal birth and are already comfortable using them, but a base size wrap will be much larger to pack. A Moby is a good option as well for the same reason, or a properly-fitted k'tan. A one-shouldered carry is also less of a concern when baby probably clocks in at less than 10 lbs (though mine was close to above it!). The only carrier I really would not recommend for hospital or very early use is a soft-structured carrier. Even with an infant insert, they are not ideal for very new babies and would not be as comfortable immediately postpartum as some other options.
Having a c-section? You definitely won't want anything around your tummy! Some said that they still found a Moby comfy, but most said that either a ring sling or a k'tan was most comfortable while healing from surgery, as there was nothing on their waist at all.
As always, only do what you are comfortable with. Only wear when you and baby are ready, and consult your doctor and/or baby's doctor if you have any concerns. Also remember that newborns are more at risk for breathing difficulty since they have no head control, so be sure you are comfortable with your carrier before putting baby in. And we are always happy to help in the chatter group if you post photos to troubleshoot!