It is true that trains in india get booked much in advance... but u would not need much of train journey if u plan it accordingly.
The golden triangle area can be done by road only as the Delhi--agra-jaipur are separated from each other by around 4 hrs by road.
Then u can also visit Ajmer, pushkar which are all nearby areas.
From Jaipur or any other city in Rajasthan u can take the ariel route to Mumbai or take the train if its available. Jaipur to Mumbai has many low cost flights.
Once u reach mumbai u can go to Goa by bus also, it is an overnight jouney. then again goa to Bangalore is an overnight journey in a luxury bus...
Bangalore---mysore---ooty are again separated by 4 hrs each by road.
Do let me know if u need any more info
Have a nice trip