If snapping is on, maybe there's something else that the new piece is snapping to? A screenshot could be helpful here.

when building, it can sometimes be necessary to use a piece of wood to create a snapping point for a piece you wanna drop. this method allows you to do lots of crazy things so try it out

So, I built a stair thing. Then I tried attaching the floor, but it hung over the front of the steps. I then added a door and tried from that and it sort of worked but then all of them were overlapping and not 'clicking' together.

try pointing your cursor along the edge of the floor you're building on. that will have the new piece snap to that edge. don't hold shift, it shuts off snapping (allowing you to place pieces more freely).
if you can't see the edge of the floor you want to snap to, (for instance building a bridge) you can place the second floor on top of the first, so it's on top and over hanging the first some. then place a 3rd. if you do it right, the 3rd should snap to the 1st correctly and you can delete the 2nd.

Not sure what is really happening with your situation, but when I use the "Left Shift" button, it is so I can place something like the floor piece in the middle of the wall instead of the actual snap-to location. Without use the Left Shift button, you have to get the cursor "close" to the snap-to part of the object, in your case, this would be the edge of the floor piece or the end of a beam. This can be a pain sometimes depending on were you are standing and your view angle. At least that has been my experience.