In The Legend of Korra, we only see Republic City, the Spirit World, small villages, etc. but what about he big cities in The Last Airbender. Does Omashu, Ba Sing Se, and other cities still exist?
Yes, many of the big cities still seem to exist, they're just not narratively important. Given that decades have passed since the previous series, it's possible that some have suffered disasters or shrunk or any number of things, but there are references for several of the major cities still existing in some form.
Omashu: In the episode "Reunion," we hear reference to Omashu in the train station P.A.:
"The 4:15 bound for Omashu is now leaving on track number six. All aboard!"
Ba Sing Se:
Ba Sing Se, as the head of the Earth Kingdom, actually features in several episodes, particularly "The Earth Queen", in which Mako and Bolin visit family who live there while Korra and the rest of the cast search for reported Airbenders in the city.