Right now I'm playing Batman Arkham City and Skyrim with xBox 360 controller, since I am annoyed by clunky mouse movement. But probably for Diablo 3, I think I have to use mouse, so I wonder if a gaming mouse can improve the playability significantly compared to the generic logitec optical mouse that I have. If so, what mouse do you recommend? My budget is pretty low, let's say below $40.
Well, if you define gaming mice as a mouse designed around a very stable sensor, then it can have some tangible benefit to your accuracy and so on. If you define a gaming mouse as a mouse with a lot of buttons and a fairly common-grade sensor (the razer naga for instance) then the benefit is more or less limited to whether you needed 12 buttons to keybind on your right hand.
Under 40 dollars the best mouse that i can think of is the Logitech G400, from a technical perspective. It has a couple of thumb buttons which are more convenient in some genres than other, but most importantly it has a very accurate sensor that responds uniformly every time that you move it (this helps make certain motions come down to muscle memory more so than onscreen visual execution as you get used to a piece of hardware).
Diablo probably isn't a game that will really highlight the technical benefits of any mouse though, and it's pretty short on keybinds. If you want to spend 40 dollars, a G400 might be a good way to do it. If you'd rather not spend 40 dollars, I don't think it's going to significantly hinder your experience in Diablo. It would be different if you were asking about a PC shooter game or a very fast RTS or something. Hope this helps.