Whenever I connect my Samsung Galaxy S6 to the wifi at home, everything else connected to the internet begins to have connectivity issues This isn't exclusive to one wifi network either, it's happened at three different locations using three different sources of wifi. 

If I connect my phone to the wifi, the computer and other phones and laptops connected to the same wifi will run slowly; videos buffering for indeterminate amounts of time or not at all, web pages failing to connect, ping issues while someone is playing games on the computer and also issues with packet loss (so I'm told).

I never had an issue with this while using these same wifi connections in question, until I updated my phone a couple of months ago. I couldn't definitively say the version of Android software that I updated from because I'm not really savvy with this kind of thing and I can't remember. But the version I'm running now is 7.0.
Same issue at my home, I know what's for me: my phone starts uploading photos to my Flickr account, takes a lot of upload band and, of course, download slows down; it is enough to pause Flickr transfer to go back to the usual transfer rate.
Maybe you have any App backup or sending huge amount of data on to the InterNet; you could start to check your data usage statistics on your phone.