Can you recommend some examples of llanera music to listen to?

The genre called "country music" or "country and Western music" is popular throughout the United States. It's not my favorite music so I'm not an expert. I live near Boston, Massachusetts. Boston is an East Coast city far from plains (llanos) and cowboys, but it is quite popular here and there are several radio stations devoted to it.

The "capitals" of country music in the U.S. are Nashville, Tennessee and Branson, Missouri.

Country music began as a kind of folk music of the people, but now it is a big business, recorded in studios by people who make it their career. 

I looked at some YouTube clips and Amazon mp3's of "llanera" music. It seemed to me that most or all of it used a harp. I can't think of any U.S. dance music that uses a harp.

The biggest country music stars are known to almost everybody in the United States. Johnny Cash was one of the most famous country singers of all time. "Ring of Fire" was one of his biggest hits:

https://www.youtube.com/watch?v=It7107ELQvY

There is a kind of joke that country music tends to be about certain themes... broken hearts, for example. A friend told a songwriter that his country song wasn't perfect because it didn't say anything about "mama, or trains, or trucks, or prison, or getting drunk," and the songwriter added a verse:

I was drunk the day my mom got out of prison

And I went to pick her up in the rain

But before I could get to the station in my pickup truck

She got runned over by a damned ol' train!

How many country music stations there are is a pretty good indicator of how popular country music is in that region.  Texas has a lot of country music stations.  When I'm on a road trip driving through a Southern state and trying to find a station to listen to, there's always a lot of country music stations also.  When I lived in Irvine, California, I don't recall ever hearing a country music station, but I would guess there's at least one but I just didn't notice it.

Another indicator is the musical entertainment at state fairs.  I've been to the state fairs in Dallas, Houston, and Austin and country music is a must-have whether in the main concert venue or on the smaller stages.  Sometimes there's even non-stop country music on the smaller stages; just one band going on after another.  The state fairs I've been to in the Los Angeles and Orange County area did not have a noticeable country music presence.  The small stages were mostly occupied by rock cover bands.

Regarding country music dancing, I've been to some country music dance halls in Houston and Austin that were quite fun.  Mostly it's couples dancing, and some were really good, with twirling and spinning and stuff.  Every few songs, they'll play a fast one and everybody will line up on the dance floor for some line dancing.  If you want to experience the stereotypical Texas image shown on TV, I'd recommend visiting one.  Most people are really in boots and cowboy hats dancing to country music; it's as Texas as you can get.  The one in Austin was decorated like a small town in a Western movie, kind of like how the Paris Casino in Vegas is decorated like a Parisian street, with a large dance floor in the center.  Even if you don't dance, it's fun to just people watch and take in the atmosphere.

@Miguel, you wrote: I want to compare it with the ''Country music'' in Venezuela called: Llanera.

My teacher shared with me this video from Venezuela of dancing with llanera music. https://www.youtube.com/watch?v=r7IegHERdNs

It seems a bit similar to me to folkdance/square dance with country music, for example, here: https://www.youtube.com/watch?v=AGn_T7u6U2Q

Square dancing is no longer popular with young people, but line dancing to country music is VERY popular in Kansas-- many clubs and bars have it, you can learn it even if you are drinking, you do not need a partner.  Here are some samples of modern country line dancing which is still popular in some parts of the U.S. https://www.youtube.com/watch?v=zb9HGNTfjr8

I think it depends on what part of the United States you are in.  I live in California, and there are not many country music stations on the radio here (honestly, I'm not sure if there are any.  The stations I remember hearing had: pop music, older rock music, music from Mexico, talk radio, a little jazz...I'm not sure if there are any country music stations).

If you travel to other parts of the United States, maybe you will hear a lot of country music.  Country music is generally thought to be more popular in the Southern states. 

When people answer this question, I think it will be helpful if they say what state they live in (or traveled to).

Country music is very, very popular here.  There are more radio stations that play country music than any other genre of music. 

I found this source, http://www.musicbizacademy.com/articles/radio/stations.htm.

KRTY 95.3 FM in San Jose.

KNCI 105.1 FM in Sacramento.

KUSS 95.7 FM in San Diego.

KZLA 93.9 FM in Los Angeles.