The city of Boone itself is not particularly remarkable IMO. 3 hrs drive SW is Asheville, a little better than Boone, partly due to the impressive Biltmore House, worth seeing.
October is fall "leaf" season in NC/TN/VA mtns, 2nd-3rd weeks receive peak tourism visits, along the BR Parkway, Boone, Blowing Rock, etc. You're visiting at perhaps the worst time due to high traffic. If your idea of nice riding is mixing it up with thousands of distracted motorists and RVs on a shoulderless, narrow, winding 45mph road, then head for the Parkway on a weekend from Oct 7-Nov 1.
US321 is a major corridor for traffic from the east, and most of the population of NC, so Boone/Blowing Rock always has more traffic than surrounding area, and October is the worst. BR Parkway traffic thins out as you get way from B/BR, then picks back up near Asheville, thins out again until another huge crush of traffic at Cherokee/Gatlinburg.
If you head out on the Parkway early in the AM on a weekday, even in "leaf" season, it's not too bad. Beware that cycling on the BRP means you're climbing ~90% of the time - the climb takes 30 mins, the descent takes 5, etc. The BRP is relatively flat around Boone and NE of Boone. The Viaduct section between 321 and Grandfather entrance is especially pretty, and especially crowded.
If you google you'll discover Boone does have shop and club rides with routes mapped. They tend to ride little back roads you're unlikely to discover in a short visit, naturally. I suspect there are some good rides generally North of Boone on little winding roads.
If you're a hiker, I can recommend the Grandfather Trail, which traverses Gfather mtn, 6.5 miles/ 5 hrs. Most tourists never leave the pavement so this trail will not be too crowded even then. Very nice views from this trail, take a camera. It's easier starting at the Swinging Bridge parking lot end (saves 1700' climb), and if someone can shuttle you up and drop you off, then they'll refund your entrance fee at the bottom gate, so it'll be free instead of $15 a head or whatever. Be sure to tell them what you're doing when you enter. Trail ends on the Parkway.