I asked Nashville visitors what they recommend doing beyond Broadway, and the responses were overwhelmingly helpful.

Head to Franklin
Franklin came up more than anywhere else. It's about 30 minutes south of downtown Nashville, and there's a lot to see there.
The town has a rich Civil War history — the Battle of Franklin lasted just one day but resulted in 10,000 casualties, including six generals. You can take walking tours through the historic downtown, visit battlefields, and tour homes that each have their own war story to tell.
Beyond the history, Franklin has great shops and restaurants in the downtown area. Several visitors also mentioned Leiper's Fork, a small artsy community near Franklin that's worth exploring.

Historic Sites and Museums
The Hermitage (President Andrew Jackson's home) was mentioned frequently. So was Belle Meade Plantation, which has tours about the property's thoroughbred horse breeding history plus a winery.
The Parthenon in Centennial Park is an exact replica of the one in Greece. It's free to walk around the park, though there's a small fee to go inside.
The Tennessee State Museum is free and located near the Farmers Market downtown, where you can grab food and browse small shops.

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Even if you're not staying there, the Opryland Hotel is worth visiting. You can walk through for free — just park at the Opry Mills mall and enter through the back of the hotel.
The hotel has two massive indoor gardens (The Cascades and The Delta) with a fountain and light show at 6pm and 9pm. There are restaurants inside the gardens and shops to browse.
The whole Music Valley area around Opryland has attractions like Cooters (the Dukes of Hazzard museum), the Willie Nelson Museum, and vintage shops in the strip mall behind Fiddler's Inn. Most music venues in this area play classic country if that's your thing.

Songwriter Shows
If you want live music without the Broadway scene, check out songwriter shows. These are intimate performances where you hear hit songs performed by the people who actually wrote them, plus the stories behind the songs.
The Bluebird Cafe in Green Hills is the most famous spot for this. The Listening Room Cafe is a short walk from Broadway and offers the same songwriter-in-the-round format. Backstage Nashville at 3rd & Lindsley does daytime hit songwriter shows on Saturdays — it's all ages, seated, with a full bar and restaurant.

Unique Experiences
Cheekwood Botanical Gardens gets mentioned a lot, especially for something peaceful and beautiful. The Nashville Zoo also came up as a great option.
For something different, consider Fisher's Off-Road Tours — it's about an hour south of Nashville and they offer side-by-side tours through the Tennessee backcountry. They even provide roundtrip transportation from Nashville.
Jack Daniel's Distillery in Lynchburg is about 90 minutes away. It's a bit of a drive, but you can tour the distillery and walk around the small town.
Nelson's Green Brier Distillery in Nashville offers tours and tastings. There are several other distilleries and a winery in the same building, plus shops and restaurants.

Arts and Entertainment
The Frist Art Museum downtown has rotating exhibits. The National Museum of African American Music is another option if you want to learn more about music history.
For something quirky, there's a vaudeville-style show called The Shiners — it features singing, dancing, and aerial performers (all clothed). Tickets run around $50 per person with a cash bar inside.
Patterson House is a speakeasy bar that's not on Broadway. Category 10 is a fun bar that's better visited during the day on weekends when it's less crazy.
Day Trip Options
Arrington Winery is about 30 minutes from Nashville and offers a nice change of scenery.
Mount Juliet (exit 226 off I-40 East) has shopping and is a quick drive from downtown.
And yes — Buc-ee's in Crossville is worth the stop for that brisket sandwich everyone raves about.