Yes, you are correct, the faster the shutter the less exposure time the sensor has to light, so to keep the same level of exposure when using a faster shutter you have to compensate in some other way, either be making the aperture larger (smaller number) or by increasing ISO.
As far as shooting notes go...all modern digital cameras, like your 70D, embed the settings used into metadata called EXIF data that is part of your image file. If you press the "info" button a couple of times while looking at the pics in your camera it will show you all of the settings along with a histogram. If you are using windows, you can also right click and go to properties on an image file from your camera, then click on the details tab.
No need to keep shooting notes unless your class requires it, the camera does it all for you for every image!