it really depends, do you guys
have a console or a gaming PC?
if you have a console you can record
just the game by itself (at least you can on ps4) and then upload it to youtube but
you can't add your face to it. if you wanna do that you'll need a video capture device like the Elgato hd60 for example
now if you have a PC you can record the games with a program (software) and your face with a webcam like the Logitech c920
hopefully, my explanation is not super confusing...
in summary recording with a PC is easier, however buying a PC that is capable of recording
and gaming at the same time could be pricey (we are talking over $500 here) and you also will have to build it on your own as prebuild PC's are extremely expensive and not very good performers...
now with console's you don't need to worry about ensembling it on your own, its basically plug and play but you have limited options. sure if you have a headset with a mic you can record the game and your voice and then immediately upload it to youtube but if you want to add someone's face while playing or do a small cut/edit you'll need a PC anyways.
now about the cost...
A PC by
itself (no monitor or keyboard and mouse included)
could cost you over $500
you'll then need a webcam wich is another $100+ but the pros are that you can edit/cut your footage the way you want and all the files (webcam and game) end in the same place (your hard drive)
a console is easy to set up and can record just the game footage and then upload it to your youtube channel immediately after but as I said
if you want to add a face to your recordings you'll need:
a camera (the price varies but quality one start at $300)
a capture device as the one I mentioned (im not saying buy the one I mentioned, that one is just for reference but those type of devices start at $100)
and a PC to add/edit and render the footage.
hope it helps