Well, basically all LaTeX editors are going to handle building the dvi or pdf for you and I never use code completion. What I mainly look for is an intuitive GUI, customizability and easy integration with external programs.
On the Mac I mostly use texmaker. TeXShop is fine for quick jobs, but it lacks the power and customizability of texmaker, e.g. I want to have more control over which pdf viewer to use and how to configure it. Sometimes I use Aquamacs Emacs when I need a bit more power, e.g. texmaker only does soft wordwrap but I find hard word wrap to be more useful when you are using version control systems. Aquamacs is the best Mac version of Emacs for LaTeXing because it has all of the popular LaTeX modes and extensions built in and well-configured by default. This saves a lot of time compared to setting up plain Emacs. Your pdf viewer is also an important part of your LaTeXing setup and I like Skim for its annotation features and easy integration with Aquamacs Emacs.
On windows I use texmaker or TeXnicCenter. The latter is again due to texmaker's hard word wrap blind spot.
On Linux I generally use Kile or some flavor of Emacs, but I only use Emacs if it is a machine that I use regularly enough to go through the pain of setting it up to work with LaTeX properly. I could use texmaker on Linux too, but since it is basically a slightly inferior clone of Kile there is not much point.