A friend of mine recently had a baby (perhaps I should clarify that she still has the baby) and asked me about recommendations for a device for capturing videos. I've included my response here in case anyone else is interested.
I would actually recommend using a phone (iPhone or Android), as they say the best camera is the one you have with you. Recent smart phones have very good cameras, you just need to remember to hold if horizontally if you'll want the videos to look good on a TV at some point. The same thing applies if you want to shoot video with a tablet.
You can also get a mini smartphone tripod from a dollar store, or an inexpensive tablet tripod adapter (and tripod) on Amazon.
The biggest issue will be storage space, even if you have a phone with 64 GB or larger of space. You'll need to develop a habit of plugging your phone into your computer and moving the videos (and photos) off of the phone onto the computer. I'd also recommend installing "Google Backup and Sync" (or Amazon Prime Photos) on your computer so that it is pushing all of your pictures and videos to an online backup location. This also makes them easier to share.
If you are looking for a traditional digital video camera, I kind of like the JVC Everio HD cameras, you may be able to get one on Kijiji for about $50 or $100. They do use a strange video file format, but that's not a big issue.
Hopefully that's enough to get you started, but feel free to ask for clarification at any point.