There are many eras and styles of country music, and each has characteristic rhythms, instrumentation and techniques.
The Origin: Ralph Peer goes to Bristol, TN, to record local musicians, including Jimmie Rodgers and the Carter Family. Instrumentation is acoustic instruments, like acoustic guitar, autoharp, and fiddle (books have been written about old-time fiddle styles), and close harmonies, with repertoire largely collected from local communities. Mother Maybelle created the Carter Scratch, where the melody is played on the low strings, with harmony and a snare-like rhythm on the higher strings. Jimmie Rodgers is essentially a blues singer with additional yodelling.
Western Swing: The elements are in the name -- "Western" pointing to silver screen singing cowboys and "Swing" coming from white musicians trying to play jazz. You're seeing the entry of pianos and drum kits, archtop guitars playing Freddie Green chords, fiddles trying to play sax leads and console steel guitar doing the horn sections. Bob Wills and the Texas Playboys are the basis, and currently the standard is Asleep At The Wheel.
Bluegrass: The last purely acoustic genre in country. Bill Monroe formed the Bluegrass Boys, playing Peer-era during band music at faster tempos and higher keys (B, high for a male singer, for the "High Lonesome Sound), with mandolin, acoustic guitar, fiddle, Dobro, and acoustic bass.
Honky Tonk: In the 40s, Swing got boiled down to the essence, with less emphasis on virtuosity, and it was called Jump Blues. In Country the same thing was happening, and that was Honky Tonk. This is where the electric guitar started coming in, and the beginnings of pedal steel. Hank Williams is the core here.
Rockabilly: Instead of "Country", the genre was called "Hillbilly", and if you follow "Rocket 88" into the future, you get Rockabilly. Small bands with each player playing as much as possible. Elvis Presley, Carl Perkins, Jerry Lee Lewis. Electric guitar with tape delay giving "slapback", acoustic bassists getting percussion by slapping strings against the fingerboards.
Countrypolitan: I love Chet Atkins as a player, but as producer and label head, we get strings instead of fiddles, smooth instead of rocking, and the Jordanaires pouring syrupy harmonies behind the singers. Think Patsy Cline for the best side, and post-Army Elvis for the part I hate but you might not. This is the Country that caused Willie Nelson to leave Nashville.
Bakersfield: Recorded in California not Tennessee. Lyrically, we start getting trucker songs. Musically, you have four-piece bands like the British Invasion had: guitar, guitar, electric bass and small-kit drums. Mixed to sound good on AM car radios, it's treble-heavy. This is where the Fender Telecaster became known as the Country instrument, and between this and Rockabilly, the start of the hot country guitarist. Buck Owens and Merle Haggard are the name artists here.
Outlaw/Progressive: This is where Willie Nelson ended up. They took more from rock music; I think Waylon Jennings kept a phaser pedal on through most of the 70s. It can run the gamut from way folkie songs with acoustic guitars and many verses, like "Pancho and Lefty", to full-on massive productions like "Devil Went Down To Georgia" by Charlie Daniels.
Southern Rock: I don't consider this to be a Country genre, but the influence, more from Lynyrd Skynyrd side than the Allman Brothers side, but that lead into 38 Special, which lead to Alabama. Full electric 70s stadium rock lineup.
Country Rock: Similar time as Southern Rock, but different influences. Instead of the big rock sound, you're getting more counter-culture sensibilities into the post-Honkytonk country. Large overlap with the Laurel Canyon Singer/Songwriters. Later Byrds with Clarence White on guitar, Gram Parsons and the Flying Burrito Brothers, and especially the Eagles, who have a great influence on Country up to today.
New Traditionalists: There was Urban Cowboy, which was kinda Countrypolitan for the 80s, but I have nothing to say about it, but the backlash gave us people who wanted to do new things with the Bluegrass and Bakersfield sounds they grew up with. Dwight Yoakam, Ricky Skaggs, Lyle Lovett, Alan Jackson and the greatest, George Strait.
Alt-Country: At first pass, it was Uncle Tupelo, the Jayhawks and Whiskeytown, and by and large, the boundaries were formed by Neil Young's Harvest, REM's work on IRS esp Murmur and Neil Young's Rust Never Sleeps, but eventually some music that actually had Country bona fides became involved, like the Old 97s. Freakwater and New Trad artist John Anderson both recorded "Wild and Blue".
This gets us to the 1990s, leaving Garth Brooks, bro country and a few other trends that, to me, don't rise to as much significance for discussion.
In General: Country is social/dance music, with simple rhythms and tempos that anyone can move to. Unlike rock, jazz and r&b, you're likely to hear waltz-time songs from country in most of the subgenres mentioned. I think you can live a long life without hearing 5/4 or 7/4, though.
The point to Country is largely the lyrics and singing, and as mentioned, dancing. "Three Chords and the Truth" is a phrase associated. You'll hear guitar (acoustic, electric, steel), mandolin, fiddle, drums (a lot of train rhythms on the snare) and singing. Accordion, esp from Texas bands with Tejano influence. Piano, maybe organ, but synths are rare and not front and center. And, while you might think what you hear on country radio is uninspiring, there are some great players playing the sessions.
The Three Chords with the Truth are not necessarily the I, IV and V, although that's common. A song that Buddy Miller loves to sing, Tom T. Hall's "How I Got To Memphis", is I ii IV, at least in the verse. And that's diatonic and major. You can write sad songs in a major scale; saddest song I can name is "Buenos Niches from a Lonely Room" by Dwight Yoakam, and it's in E major.
Also, there has always been an interaction with black popular music in country, where there's a draw and rejection, from DeFord Bailey to Ray Charles to Lil Nas X.