I truly believe a good cruise travel agent is worth their weight in gold. Its best to have someone you can call with your questions or concerns and that will get back to you in a reasonable amount of time. In many cases, they offer some group pricing that by booking direct can't. If for some reason the price of your cruise does lower in price, they can get you credits or the lower price. Just be sure to find one that does specialize in cruises.
You will probably get the best deal on the cruise portion of your trip directly from NCL. However, flights and any pre/post-cruise stays would probably be cheaper through your TA. And as Trip and Donna mention, your TA can be your advocate if there are any issues.
A Bad Day At Sea [with power] Always Beats A Good Day At Work
Coral Princess - tour starts 5/31 in Fairbanks
Yes, try both the agent and NCL. NCL pricing will change over time, so you might be able to get a good deal at a certain moment. On the other hand travel agents make group arrangements with cruise line companies and might be able to offer you something sharper. For tips on how to get the best deal, ***edited to remove commercial reference***.
On an NCL cruise you can and will get the best deal from a good travel agent.
A good agent can discount their commission on an NCL cruise to get you a lower rate than you can get directly from the cruise line. Agents cannot discount on RCI cruises nor can they advertise a lower price on a Carnival cruise but can offer a lower rate if they are personally contacted by the client.
As others have said a good TA will be in your corner when things go wrong. If you book through the cruise line they are as interested or more in protecting their own interest as in protecting yours.
Here is a good article by Kuki on how to find the best price on a cruise.
At the very least a travel agent will be able to at least get you the same price as the online price. If they are significantly more, chances are the agent is mainly worried about their commission. Also, some agents will throw in little extras when possible.