Sorry, it seemed like you were another one of the people that come on here trying to make the next great thing with no experience. To go about starting it?

I learned C++
I started programming a few hours every day
I found a graphics library and started programming
I released my first API!

The best way to get started is to, well, start. Learn a programming language! Although I started with C++, it would be far better for you to develop in C#. Some good tips:
1. Get Visual C# Express 2010 (It's Free!)
2. Go through their Coding4Fun tutorials and get a firm understanding of C# in general. I learn best by modifying the examples and reproducing the code myself. Challenge yourself to find practical applications for everything you learn, or it'll fly right over your head.
3. Try some graphics programming! Find a library (I'm sorry, but you'll have to ask someone else for a recommendation ) and get coding!
4. Start off simple, with Pong (Holy Crap, I felt so good finishing it. Pong is not stupid.), or tic-tac-toe
5. PROGRAM! The reason I didn't learn: I wasn't programming. Program all the time. Just start programming!
6. Once you have a group of good, well-programmed games under your belt, learn Unity3d.

Unity3d uses C#, and has a great documentation. Use it, make some games, experiment. Once you become familiar with programming C# using Untiy3d, join a team or create one and make your dream game! If you've got this far, and have enough games to show, you can attract artists and programmers, and you can make your dream game. You have a long road ahead of you, but by all means, I don't want to discourage you, I'm just telling you to take it slow.

You are getting strange answers because your questions don't make any sense.

Anime style graphics or any graphical style is not a feature of any engine. It has nothing to do with an engine. It's like asking what breed of dog can best help you do your history homework. One thing has nothing to do with the other!

An engine is hard to define because the term gets thrown around so much. In general, an engine is something that handles all the low, system level details, and acts as a base layer that a game can be built on top of. You create scenes composed of 3d models, animations, sounds, custom behavior scripts, and whatever else and the engine takes care of getting them all played back for you.

People use engines because they are known, stable, reliable, frameworks they can build on instead of spending lots of time and lots of money developing their own. The better engines also run on many platforms, and have fallbacks to handle all kinds of different setups. So it doesn't matter what engine that is running that game. The interesting part is the actual content. The artwork, the sounds, the programming (game rules), and whatever else.

As for the animation style, this is created by the artist(s) in their 3d modeling and animation software. They are polygonal 3D figures with simple texture maps. They are being lit with a standard technique called cel shading, that mimics the way traditional cel animation is drawn. There are hard boundaries between the highlights, midtones, and shadows instead of a nice, smooth interpolation.

Creating these types of graphics, and the manga/anime style is an art thing, and has nothing to do with programming or engines. You must first understand art, figure drawing, and then the rules that make manga and anime style what it is.

You can try the books of Christopher Hart.
https://play.google.com/store/search?q=christopher+hart&c=books <- Go read all the free previews!

Then for the 3D part, grab a copy of Blender3D (free), and watch these video tutorials. They cover Blender basics, and then teach you how to get started making a simple anime / chibi character.
http://www.youtube.com/user/PXstriker/videos?flow=grid&view=0 <--Peter's massive Blender Tutorial

Newbie aspiring game developers reading this (I am one myself, so I will keep this simple and gentle as possible. ) :

With more than two years as a 2D and 3D artist in the industry, I can assure you that for most AAA popular games, the art assets alone took hundreds or thousands of labor hours of work, not including the programming and other needs.

I am taking the start simple and work step-by-step approach which almost every experienced game developer says is necessary.

Rearch and develop each simple game thoroughly before moving to the next one, making sure that you fully understand the whole code. If you don't want to code games as a game developer or working for one, then you may become a game maker which leads to becoming a game designer using pre-made tools.

Whatever you do, research, work hard, and have fun, fun fun!!


Clinton

Thank you all very much for your input and advice. Superman3275, thank you for your tutorial, I'm glad we could clear up our confusion. My questions have been based around a basic lack of understanding regarding this kind of work. I have a project that I mean to see through to the end, one way or another, so I thank you for all your help. As I and my team progress in our understanding of basic concepts, I hope to return to you with questions that are more informed, and make more sense, as so to avoid previous, shall we say... awkward misunderstandings. I am sure I will return with more questions, as I've gotten more educated responses than I expected to get with such a broad question. Again, thank you all, for now, goodbye.

-Rowan

Step 1
I would strongly suggest finding an engine first, that will have the necessary programming language for your team. Don't learn a language first, use your teams strengths.

Step 2
Make a very simple game to get your team working well together before taking on a large project like this.

Step 3
You will know all you need at this point to comprehend the massive scale of the project your team wish to create. Good luck.