All different kinds are suitable from newborn! And yes you can wear it over a coat or if it is cols pop it on under your coat with the baby just in a sleepsuit and do your coat up over the both of you :)
A lot of people start with a stretchy wrap for a newborn- brands like kari me, moby or victoria the sling lady. Or a Close carrier which is kind of like a pre tied wrap, is adjustable and bit easier if you don't fancy wrapping.
Woven wraps will last from newborn to whenever as they are nice and sturdy. They are bit harder to get used to than a stretchy but are great.
Or a more structured carrier like a baby hawk Mei Tai is a square of fabric with straps you tie it on with. Again suitable from newborn to toddler.
Or if that all sounds very complicated you can get carriers which are full buckle so just as easy to put on as your high street carriers like baby bjorns but unlike baby bjorns are comfortable, hold the baby in optimum position and will last for years as you can wear teh baby on your front and then your back as they get older. Ergos are one such brand, boba, beco, connecta and so on. SOme are more structured than others.
If that all sounds complicated then have a google to see if there is a sling library near you where you can try them on and hire them out to see what you like best before you buy :)
Most are available per loved easily enough, the FB group Slings and things- FSOT is a very active sales board. Don't be put off by the expensive wrapping habits they all have, you will also find lots of help getting more reasonable sling there.