Ok great....now I get what you are more into.
Yes a Kerala - Goa west coast theme sounds really good. Be sure to stop by Mangalore on the way between Kerala and Goa at some point.
Of the major cities you suggest in south....each one has pros and cons...but I think for your format of travel...Bangalore would probably be the easiest to visit since its closest to the west coast region you are already touring and you can easily take a train or bus to it from say calicut in northern kerala (kozhikode as its called now). You can even stop by the historic city of Mysore (the famed Tipu Sultan's capital) in a circuit back to the west coast. Let me give you a map of what I mean:
So you can do your thing in Kerala....very well developed boat based and cultural tourism in many areas.
@Levina can give you some input on what is best for there in say a week of spending time.
Then if you swing through Coimbatore (my hometown) (en route to Bangalore)...you can check out the Nilgiri hills for a cpl days or so which is a mountainous scenic area (with lots of old british houses and such) with its own special heritage railway steam train:
After which you can proceed on to Bangalore (Bengaluru these days) which I can give some details about later (after you have a more firm overall plan in mind after reading this)
From Bangalore you can then go to Mysore (Mysuru) which has plenty of cultural things to see (Mysore palace, tipus summer palace, lots of temples and historic buildings and various parks and gardens etc etc)
Then you can swing back to west coast to Mangalore and proceed on to Goa and finish your time there.
This second leg of the journey I think can cover one whole week...and can extend it to two if you want a bit more leisure unrushed travel etc.
For strict 2 week travel I might suggest skipping the internal detour and just keep it a west coast affair for simplicity sake so you are not rushed in finding more places to stay while you are touring etc (i.e just connect to mangalore directly from calicut and skip out the bangalore thing)....you can add just mysore to it as a semi-compromise since its actually quite a nice cultural place.