"We really want to visit this part of the country... A mix of touristy, non-touristy and offbeat would be great for us."
Freiburg itself isn't very "Black Foresty" and Titisee is among the most touristy places.
I suggest you start with a visit to the Black Forest Vogtsbauernhof open-air museum in Gutach (Gutach Freilichtmuseum station is at the doorstep.) This place gives you the best one-stop introduction to local life and culture.
From there you might want to ride the scenic Black Forest Railway line up and into the more mountainous central part of the Black Forest as far as the town of St. Georgen (station with the highest elevation.) This ride was an engineering marvel at the time of its construction and one of Germany's most scenic routes. There's nothing interesting to speak of in St. Georgen - take the next train back from there.
Map - Gutach Freilichtmuseum stop is new and not shown, but it's near Hausach: https://upload.wikimedia.org/wikipedia/commons/4/4d/Verlaufskarte_Schwarzwaldbahn_(Baden).png
Stop in Triberg on the way back if you wish but it's a very touristy town.
Haslach is a very handsome and nice normal town on the BF Railway with a museum that displays local traditional costumes. Just outside town there are a couple offbeat things - a Nazi work camp memorial site - Gedenkstätte Vulkan - and an old silver mine (Segen Gottes.) (Try google translate feature on this page.) You might need a taxi for these last two.
An even more attractive half-timbered town is Schiltach - just off the Black Forest Railway on an offshoot railway from Hausach.
On the way back to Heidelberg you will also pass through the old walled town of Gengenbach - might be a nice spot for an evening meal. This photo of the town captured a wine fest.
This is an over-full plate for one day, even a very long day, so you need to pick and choose what looks best for you. But I think you'll get a better idea of what the Black Forest is about by taking this route.