I agree with what Jan said. I too have been on the Victory twice (once last summer from San Juan) and I have been on the Mariner of the Seas too. So I have experience with both.
You will hear different opinions from different people as to which line is "better". There are a lot of Royal Caribbean enthusiasts who go around and bash Carnival and the reverse is true also. Many of the people who will tell you not to sail on Carnival have never even tried Carnival (or if they have sailed on Carnival it was years ago when they tailored to a younger crowd.)
To be honest with you they are very similar products. Royal Caribbean ships are bigger, but their standard cabins are smaller. The food quality is debatable since what is good food for some great food to others. Since this is your first cruise, you won't know any different, so why worry as to which is better than the other? In fact most of the time, the cruise line you pick for your first cruise will become the one you prefer since you will compare any other to it. Chose the cruise that is in your budget, don't base your choice on "they say" advice. You will have a great time on either. If you find you enjoy cruising, you can always try the other line for your next cruise.
As you can see from our cruise choices, we are Carnival people, we tried Royal Caribbean and it was great too, but we did not see enough differences to warrant the extra costs. Now, there are some cruises where RCCL has less expensive fares but for the newer bigger ships, I have not seen it. Pick the one you can afford, the savings will help pay for your on board expenses.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007