well im not sure if you meant you had 3 week or 3 days to get overland from bangalore to mumbai but im going to throw in the name of another place anyways
is a hill station not far from mumbai and is very peaceful and theres some nice short walks to do. there is also a small toy train that takes you up there but in jan when i went it was shut due to monsoon flooding and had not at that time been repaired...
not yet been to hampi (hospet is the station) bit its meant to be fantastic (just take lots of mozzy spray is the advice ive been given)
with reguards to booking your train then seats into and out of mumbai get booked up very quickly so i would advise you to book your seat as soon as possible, the tourist quota gets full up very quickly, my next three train journeys in india are all standard class because the tourist quota is already full...
also book a room in mumbai as soon as possible because again they fill up very quickly, my partner and i rang every accomm in LP and RG and all were full apart from one who said he 'might have a room' luckily a girl checked out as we arrived so we got the room... and i when i say we rang every acomm, im talking from budget-top end not just the budget ones!