What's the best website to send big files that are too big to send via email that is free or cheap
+ 1 for these.iCloud Drive, Google Drive, DropBox, Box and the likes are a good way to share big files.
If it's attachments that are big, use Mail Drop. You can send attachments up to 5 gb in size. I've used it a few times, and it works great. Here's more:
https://support.apple.com/kb/ph2629?locale=en_US
Is that like AirDrop for photos? I've never used maildrop. Does it go directly a email address of your choice? Do I have to set up a iCloud email address to use this