Teach me how to buy a TV for video games

I don't play any modern games and in my whole life I've only owned CRTs. All I know is in the mid 2000s HDTVs were considered the devil for games because of "omg teh display lag". But now it's gone on for long enough that I can't buy any current systems or games because I lack a HDTV. I would like to finally buy a PS4 (and later down the line a Switch).

I don't know what to look for or how. Please assist me so I can join you guys playing in the 2010s.