advise me on what route to take
 

cheapest way to travel from Trocadero Citadines apt to Versailles palace would be by cab or metro? and how far is it?

Versailles is in a suburb of Paris and the Metro doesn't go there at all - you have to take a regular train. I beleive the RER is aobut 3 euros per person to get to Versailles. Obviously a cab would be much more than that - but value would depend on how many people are going.

The RER C that goes straight to Versailles follows the Seine on the other side. If you are really at Trocadéro (not automatically true in spite of the Citadines name), just cross the bridge to the Eiffel Tower and catch the RER from the station that you will find slightly to the right: Champ de Mars-Tour Eiffel.

well we are 9 people.

how much would it cost if we grab a cab??

You won't get 9 people in a cab! You'll need to get a car service with a mini van. I checked one service and it's E239 for 8 seats for 4 hours and E410 for 8 hours. 4 hours is not enough time to get to Versailles and back and see something of the chateau.

Take the RER - it's very easy. Buy your ticket, get on the train to Versailles Rive Gauche (15 minutes). Get off the train and follow the crowd.

can you tell me how much would be the ticket fare? and 15 minutes from where? as i said i would be in Trocadero..

The fare is 3.05€ one way from anywhere in Paris to Versailles-RG.

Here are some travel times from Champ de Mars-Tour Eiffel, just as an example:
Départ 09h30 09h45 10h00 10h15 10h30 10h45 11h00 11h15 11h30 11h45 12h00
Arrivée 09h57 10h12 10h27 10h42 10h57 11h12 11h27 11h42 11h57 12h12 12h27

thank you. what would be the nearest station from Trocadero?

You're not paying attention to what people are telling you!

nytraveler gave you the price.

kerouac gave you the directions.

Here it is again.

The price is E2.9 each way.

You can walk across the river and take the RER C at Champs de Mars.

If you don't want to walk you can take the metro (I'm guessing you gave me the metro station you're at, not just the general area) from Trocadero - line 4 in the direction of Porte d'Orleans. Get off at Champs de Mars/Tour Eiffel. Exit the metro. Walk across the street and buy a ticket for Versailles. Get on the train. Just follow the crowd.

It is 15 minutes on the RER from Paris to Verasilles Rive Gauche, just as I said above. Make sure you get on the train for Rive Gauche.

Do not lose your tickets as you need them to exit the station in Versailles.

Thank you very much everyone!!!!
I'm gonna post some more questions about my trip. Guide me for the same. Thank you once again