Doors can be triggered to open, and there are a number of ways you can do that. First, you can create a brush, covered in the trigger texture (Just search for "trigger" in the texture browser), and tie it to a trigger_multiple entity. Be sure the door is named, and in the trigger's Outputs tab, add and output that targets the door and, "OnTriggered," opens the door. When you walk through the trigger, the door will open. The same could be done with a func_button entity, as well.
You can also make the door open with the "Use" key by checking the "Use Opens" flag in the door's properties' flags tab.
To get the door to open by simply walking into it, go to the door's properties, swap to the flags tab, and look for a flag that reads "Touch Opens." This is usually on by default, so I find it weird that it didn't work for you on the first try.