1

I`d go to Bangalore for a couple of reasons. Mumbai`s airport just isn`t a "civilized" experience, and you are likely to find more in the budget line of accommodation there.

Bangalore to Hampi is easy. Goa-Hampi-Goa has become a stupid trip - full trains and crappy buses.

2

It would be interesting to know your current situation.

I assume (probably like Pirate - I endorse his idea that you fly into B) that you have not bought your ticket and visiting India from overseas - in which case there are better options than flying into either M or B if your intention is principally to holiday in Goa with a side trip to Hampi. I suggest you look at flights into Goa transiting thru' the Gulf on Etihad, Emrates or Qatar.
Since you are short of time you could choose an openjaw ticket via the Gulf into Bangalore and out thru' Goa - I've done this 3 times from the UK and it works well.

Altho the train Goa - Hospet (for Hampi) may now be crowded, there aint many options as the bus is far worse. You could take a bus from Hospet to Hassan, spending a coupladays exploring thw adjacent religous sites, then bus down the ghats to Mangalore (there is also an overnire train) and train North to Goa, there is a daily train to near Palolem.

3

In my opinion, there are far more interesting things to see between Mumbai and Goa than Bangalore and Goa.

Chiplun, Ratnagiri and Vengurla are places on the coast between Mumbai and Goa. Chiplun is not exactly on the coast, but near it is a place called Guhagar which has a beach.

4

I missed your Q wrt the coast between Mangalore and Goa - the one distinctive place is Gokarna - a pilgrimage town assocated with Hannumann, but with some idyllic bays just South of the town. That coast South of goa has many beaches, many quite isolated or alternatively popular with local families.
I suspect #3's suggestion may be suitable for just a few purists but not for an individual doing a fortnight's trip.

5

I would choose Bangalore.From there you could visit Mysoore,Hampi and from Hampi travel to Jog Falls(from Hospet to Shimoga and than Jog) and over the Ghats to coast for Gokarna and little further north to Goa.

6

From Mumbai, the places a;long the coast are Murud,Janjira,Ganapatipule,Ratnagiri,Malvan,Sawantwadi which is just 75 km from Panjim. Ganapatipule probably has the best beach. The cuisine at Malvan and Sawantwadi is interesting as also local customs.
North of Mangalore, the places along the coast are Udupi,Malpe,Marabanthe,Murudeshwar,Kundapura,Honnavar,Gokarna,Ankola,Karwar from where Panjim is about 110 km.
If you move inland from the coast Karnataka probably has more variety of forests,heritage sites, coffee plantations and scenic coastal areas.