I live in the region around Baden-Baden.
Let me first state that the previous answers are basically correct. Using bahn.de, you can find trains and busses that will take you everywhere you want, especially to the more well-known tourist locations. So there is no need to rent a car.
On the other hand, however, although Baden-Baden is not a small town, the region around Baden-Baden, especially the Black Forest, is a rural region. This means, depending on your chosen destinations, the net of public transport might be somewhat sparse, resutling in longer waiting times and cumbersome connections.
This is not meant to scare you off from public transport. As previuosly stated, it is definitely possible to use public transport only. I just want to point out that there are certainly some benefits regarding comfort and time if you rent a car. I guess it is a trade-off between comfort and cost.
Although you not directly asked for it, allow me to suggest some destinations:
The Autoroute "Schwarzwaldhochstraße" and the lake "Mummelsee" are great scenery spots and good as a starting point for hiking if you enjoy that (the weather here is comparably mild for November)
In Baden-Baden itself, I suggest to visit the Casino (you dont need to wear a tie, a classic sportsjacket or a blazer are enough)
In Bühl, the neighbouring town to Baden-Baden, you can visit the ruins of a medieval Castle (I recommend climbing the tower and enjoying the view)
Karlsruhe is the nearest major city. If you are interested in Bars or Clubs, this might be the place to go. It is also home to a well-regarded Universitiy. In the city center, you will find a palace with a historic museum inside. Also, the museum of natural history is quite good.