The amount of space consumed by recording videos is related to the resolution at which they're recorded. Lowering the resolution will use less space on your device, and speed the upload process when you upload the videos to Google Drive. This quick clip will show you how to change this setting on an iPhone. Android users should be able to make similar changes to their phones.
You can then use the Google Drive app on your device to upload your videos to Google Drive.
Even with the lower resolution, videos recorded on modern smartphones are still very high-quality recordings, and create very large files -- much larger than we need for our purposes. If your videos are more than a few minutes, the time it takes to upload the recording can be annoyingly long, especially over a slower internet connection.
There is an iPhone/iPad app called COMPRESS that can compress the size of the video file to a much more manageable size (1/3 original size). This allows you to make the file smaller, which means it can upload to Google Drive much faster. There are similar apps for Android. See example below -- notice the quality is not significantly lower, but the size is much smaller.
The first step is to make sure the Google Drive app is installed on your mobile device. This allows your device to access the files stored in Google Drive, as well as provides the ability to copy photos and videos from your device to Google Drive.
Click the links below to see an example for each device:
iPhone / iPad - Ignore the parts about "WeVideo" -- the steps for getting the photos / videos into Google Drive are all you need to know.