Aperture is one of the most fundamental and powerful tools you have at your disposal for controlling the look of your images. As such, understanding and mastering every property of it and consequence it has on your photos is crucial. This excellent video tutorial features an experienced landscape photographer giving a nuanced approach for choosing the right aperture for your images.
Coming to you from Christian Irmler - Landscape Photography, this insightful video tutorial guides you through his approach to choosing the right aperture for each of his images. It is common in the genre to simply set one's lens to as narrow an aperture as possible before diffraction starts kicking in and to adjust the shutter speed so that you can get a balanced exposure with said aperture and the lowest ISO setting. And while that will give you good images in a lot of situations, you might not be maximizing the image quality or using the optimal combination of settings for the conditions at hand. Check out the video above for the full rundown from Irmler.
And if you really want to dive into landscape photography, check out "Photographing The World 1: Landscape Photography and Post-Processing with Elia Locardi."
Since digital shots are almost free and time and location are not, it is definitely a good idea to shoot a scene at multiple apertures and focus points. Every lens has a sweet spot so that is a good starting point. If one lives in an area and can go back multiple times, that is great, but for locations where you will only have one chance, the more shots the better.
Yes, you should test your lens for the "sweet spot" where it is the sharpest as you suggest. On location, I shoot with the best aperture for sharpness, then do a focus stack as a backup. Since memory cards hold so many images, you really must take advantage of the location and time of day. You may not be able to go back, so make the best of your opportunities.