My recommendations assume you will be basically heading east across CNY (so I will not include the Adirondacks, Catskills or Hudson Valley).
Cooperstown is an elegant town with a great art museum (the Fenimore) and the Glimmerglass Opera. It is also home to the Baseball HOF.
From the Cooperstown area, I'd head west on US 20.
Taking US 20 is slower than taking the Thruway or I-88/86 westbound through the state but there are lots of towns that have what you are looking for.
You will pass through a string of rural small towns with antique shops, etc. You will eventually come to Cazenovia, another elegant small town. The stop here is nearby Chittenango Falls State Park to view the nearly 200 foot waterfall.
If you keep taking US 20, you will be entering the Finger Lakes region and will pass through Skaneatles, another elegant small town at the foot of the lake with the same name with lots of interesting shops. The next town, Auburn, has an interesting museum, The Seward House, as well as the last residence of Harriet Tubman and the last remaining Tiffany Glass chapel (Willard Chapel) in the US.
Continuing on Rt. 20, you will come to Seneca Falls, near the north end of Cayuga Lake, which has the Women's Rights National Historic Park and Elizabeth Cady Stanton House then to Geneva (at the foot of Seneca Lake), home to Belhurst Castle and Geneva on the Lake. Taking any of the routes along these lakes will take you into the wine country. There are obviously several wineries along the way.
Canandaigua is the next town along the way; it lies at the foot of the lake with the same name. An interesting stop here is Sonnenburg Gardens. From here, you could continue on US 20 west (and pass through some more "antiquing" towns and pleasant rural towns) or get on the NYS Thruway and continue west. Stops in Rochester and Buffalo (two of NYS' largest cities), situated on Lake Ontario, are worthwhile as there are MANY parks, gardens, theatres/concert venues, museums and amazing architecture in both. The George Eastman House, Stone-Tolan House and Susan B. Anthony House are worthwhile museum options in Rochester.
The Knox/Albright museum in Buffalo is a worthwhile stop.
I hope this gives you some ideas. Have a great trip!