Once upon a time I used to hike with a big digital SLR camera. Then I downgraded to a little pocket Sony, which I still use on long trips, but now I just use my phone.
Unless you’re a photographer, or photography is seriously your thing, there is very little reason to carry a big camera and gear – especially when you’re already carrying a baby. Most cell phones today have very good cameras. They can take still shots, live shots, videos, panoramic pictures, and even have filters that can be applied while taking the photo. If you’re a social media sharer, it’s way easier to share directly from your cell phone than to download then upload photos on your computer. There are even extra zoom lenses that can be attached to cell phone cameras now. And cell phones are much easier to carry and fit in the hip pocket of most carriers and packs. Most smartphones are very water resistant as well.
If you’re thing is video check out Go Pros. They can be worn, so you don’t have to carry them. I don’t currently use them, but I think it could be cool.
If you really want a separate camera but to keep it small, little digital pocket cameras are great. My little Sony was $75 at a Black Friday sale when I was in middle school and I’m still using it. Downside is I must really watch it in the rain, and I have to pre-charge and carry and extra battery. My iPhone takes better video, but my Sony has better zoom.
And if you’re going to go big, talk to a professional who does photography in the middle of nowhere. Check over reviews on weight, special considerations, gear required, ability to hold up to weather, cost, and how much you’re willing to carry with a baby.