Hey Aniwaya, welcome!
To address a few of your questions:
I would highly recommend searching for any previous builds of the item you plan to build. By checking out others' work you can find a good material that you know will work. That being said, almost any material can be utilized, but it all comes down to tools, experience, budget, and time (and more, but this is a basic outline).
Foam (eva foam) is often a nice and cheap place to start, though wood (often MDF) is another alternative if you have access to some better tools and workspace. MDF would probably be too heavy for your purposes. If you are working on a super budget and have extra time, cardboard or foam board can make awesome props, but they require quite a bit of finishing to achieve more realistic aspects (true of most materials though). Paper craft is another good approach, but also time and labor intensive if the paper is hardened with resin.
My best recommendation would be eva foam, as it is con-safe, light, and very easy to work with compared to some other materials (i.e. it is cheap and forgiving). It also requires fewer tools (sharp knife, hot glue, and sand paper will do it all, but time is your price).
I am not sure which "version" of Neptune's weapon you are attempting, so picture would help.
A word on budget:
Ask anyone here and you will get a different answer. This is largely a hobby where money is "sunk," sometimes in massive amounts, to achieve a desired goal (screen accuracy, scale, the perfect look, etc). As such, your budget for any project will most likely double or triple regardless of your initial plans (this is not always the case, but I have found it especially true if it is the first time your are attempting a process). As such, do you best to limit your scope on your first build. Since you seem set on an ambitious item, then consider simple and cheap materials that allow for error and learning. You do not want to spend hundreds of dollars on tools, materials, and other stuff to make mistakes that can cost quite a bit.
Check out any build thread you can find on items that are similar to your chosen item (guns, staffs, etc) to get a feel for others' methods, as these are the key learning tools for many hobbyist, especially for cosplayers.
I am sure many here could elaborate further, but more detail is needed.