Newbie questions re: making an ottoman

I want to try to attempt to make one of those upholstered storage cubes with the removable lid (dimensions roughly 18"x18"x18"). I live in an apartment and don't have a workshop, so I was thinking I would buy some plywood from Home Depot and have them cut it to size.
Would plywood be the best material for this? As the wooden box will be covered with batting and fabric, would just screwing or nailing the pieces together be sufficient for putting the cube together, or do I need to put some sort of joint blocks in the corners (i.e., will the plywood accept a fastener parallel to the ply)?
Any help would be appreciated.
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Well, my only other option is to get my dad to cut it, but I'm not sure that would guarantee a better result. :) Would it help to bring a template in (to Home Depot)?

What thickness would you suggest for the plywood? I don't want the cube to be too heavy, but obviously I want it to be able to withstand the weight of someone sitting on it.

Thanks for your help!
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