Okay, so a few things.
1.) Your internet speeds are only as fast as your ISP provides. This means that if you get 100Mbps download and 50Mbps upload from your provider, Cat6 cables that can do 10Gb are not going to benefit you anymore than Cat5a cables that can do 1Gb. This is especially true if the modem/router your provider has given you doesn't even have ports that support that speed. Most now days will have 1Gb ports, but unless you're an extreme enthusiast, you're not seeing 10Gb without SFP+, which I would bet almost everything you don't have.
2.) New cables are always a great purchase, especially if your's are real old. Cables can break down over time, and while unlikely, can hinder performance. The important thing is really what the cable is made out of. Do some research about shielding inside of Ethernet cables. Shielded Ethernet cables reduce noise, which could help with performance, but it certainly isn't going to give you any more speed than what you should be expecting.
3.) A VPN can theoretically give you better peering options and routing, if your ISP's suck. A lot of people do this for streaming services such as YouTube or Twitch.
Edit:
I forgot to mention that although having higher speed rated cables won't improve your Internet speed, they certainly can improve local network speed. Devices connected via your modem/router or a switch if you want to get fancy will share data at the maximum speed supported by the ports on your device. In my case, devices connected in my house all go through my 1Gb port Cisco switch, so streaming movies and music from one to the other is quick.