What is the best internet for video games?

Just no.

And assuming your asking what's the best type of connection, it's the one that's the fastest and has the biggest bandwidth - fibre optic.

If you're wondering which ISP, BT is the way to go.

And finally if you're wondering which method of connecting the router, it's wired (cat 6e Ethernet cable). It's always wired, always.

A high speed fibre optic service... not sure what you are a looking for in an answer, but you need to realise that the ISPs and services open to you depends entirely on your location and that means more than just your city (service even differs between rural and urban parts of the same town/city/etc and signal quality depends on the distance between where you live and the local exchange).