Yes, for a new baby, especially in the summer, I'd get a moby wrap or other similar. The close caboo would be sort of classed as a stretchy wrap. It is stretchy, but it's also pre-wrapped. I personally don't like this as it means you can't adjust it as much as you would a moby wrap or other wrap that has to be tied fully in the moment. Some people love them because they just don't want the hassle of having to do it all then and there and they find it easier. So that shouldn't put you off and I'd recommend trying one to see how you get on with it. But personally me, I prefer a moby so I can wrap it myself and know it fits me right. We also have an Ergo carrier, which is great, but they are best for older babies and toddlers. You can absolutely use them from birth, but they're just a bit more clunky. You also need to use an infant insert until your baby fits on their own (basically a thick duvet to wrap baby in so they will fit). This again is totally fine, but it can make it hot. If you're looking for something to use in the summer at first, I'd steer away from this as you'll probably find it's too hot with the insert in on the warmest days. We found the moby wrap was much better for the summer.
In rain or cold weather, we just wore coats that would zip up over her legs and sometimes all the way over her arms if it was really cold, and used an umbrella to keep the rain off the top. For cold when it wasn't always possible to zip her in, I'd also just dress her really warmly on the bottom, extra layer or warm leggings and extra socks. They also make moccasins for their feet that you can get to keep them warm before they wear shoes. Also if it was really raining and I had my hands full (couldn't carry an umbrella), those are the days I'd use the pushchair with the rain cover.
