I don't use a travel agent, but I am retired and love doing the research myself. I use TripAdvisor to assist on hotels, and things to do (read the reviews to determine quality). Also, I use Kayak for the lowest prices for hotels, flights and rental cars.
I suggest checking London, Paris and Rome for things to do on TripAdvisor. It will display a number of choices. Also, look for tours, just do a google search for London tours and you will find a number of choices. Rick Steves has great guidebooks that will assist as well. Kayak will show you choices of hotels by price and location. Don't book on kayak, but once you find your hotel, go to the hotel website and book the room.
All three cities have plenty to do and if you will have 3-4 days in each city. Try to plan what sights you want to see in each city. All three cities had great public transportation, however Rome will require more walking.
You can take the train from London to Paris, but may wish to fly from Paris to Rome. If you decide to take the train, consider the night train with a sleeper.
When purchasing you flight from India, see if an open jaw ticket works best, flying into London and flying out of Rome.