Just practice at holding it straight and when you frame your shot try to find a straight line you can look at. For example the water line, the side of a building etc, and if your shot isn't straight you can just straighten it up when you edit your photos.
And when it comes to opening your aperture or not it depends on the situation, but when you open your aperture you get a more blurry background. The aperture are more open at for example f/4 than at f/16.
And one google picture I found to help you understand the rule of thirds.
As you can see you try to place your objects in one of those lines.