Changing the DNS server (Domain Name Service - „The phone book of the internet“) should not have an influence on the bandwidth („speed“) or roundtrip time („ping“ time): the DNS server is only once queried, at the very beginning of the connection („give me the IP address for foo.far.baz“).
Once the IP address is known the DNS server is not required anymore. Until the next connection is made.
So yes, selecting e.g. Google‘s public DNS servers 8.8.8.8 and 4.4.4.4 can make a huge difference (compared to the DNS servers of your internet provider), especially when you‘re „browsing the internet“ (a single web page alone can query a dozen or more different internet services, each requiring a DNS call.
But I would not expect a measurable impact on online games: you get connected to some Azure data centre, maybe some reconnects in between - that‘s it.