There are 2 ways of doing this, and the method will make a BIG difference.
One option is to use something like Onedrive, GoogleDrive, Dropbox or Box. In this scenario the photos will go up in their original size and as the above posts pointed out, it will take huge amounts of space (and extra space costs $$$), would be slow and likely unpractical for really large quantities. This is a better option for backing up the originals. Option 2 is to use a photo sharing service like Flickr (even Facebook does this). When you upload to most of these services, regardless of the original file size, they will resize the image to a predefined size, usually much much smaller than the original as for online viewing you do not need a massive gazillion megapixel image. This is a better option for sharing with friends/family.
If you are just trying to have a backup, I would opt for hitting a Bestbuy and getting a 1TB WD Passport portable Hard Drive for around $100 to copy them to, than you have all the space you could possibly need and no internet speed/ file size limitations.