The main way to reduce the size of scans is by changing the resolution. This can be done at the scanner, or in the scanning software.
To print an acceptable image it only needs to be about 300dpi (dots per inch) and even 200dpi is quite acceptable. Once you go below 150dpi you will start to notice the degradation. Commercial printers usually request images at 300dpi.
This means that, if an image is printed with a size of 2", the scanned image only needs to be 600 pixels. If the print is the same size as the original, you need to scan at 300dpi. If the print is half the size of the original, you can scan at 150dpi but, if you will print at twice the original size, the scan needs to be 600dpi.
You should almost never have to scan at the printer resolution. This is because the scanner can see 16 million (or more) colours in every pixel, but most printers can only print eight colurs: the cyan, magemta and yellow inks, mixes of them (red, green and blue) plus black and white. For any other colours, the printer will put dots of those 8 colours side by side, and the eye will integrate them into a shade.
If you are scanning to view the image on the screen you only need as many pixels as are visible on the screen. In other words, for most current screens 1920 x 1080 pixels. An image that is 10" long only needs to be scanned at 192dpi. When sending photos by email, I usually reduce their size to something like 1200x800 pixels. The person recieving the email will be able to see the photo in full quality, and my email program does not have to send so much data. ALso, many email systems limit the size of an email to 10MB, so an email with just a few full-size photos might be rejected. Reducing them avoids that.
I have only talked about colour images. However, if you are scanning text or line art, you can set the scanner to Black & White or Greyscale. AT the same resolution, black & white makes the file 8 times smaller, greyscale about 3 times.
Another thing you can do is to change the file format. JPEG (JPG) files are usually the smallest. However, the more compressed a JPEG file is, the more fine detail you lose. Most cameras and scanners default to producing JPEG files, without any noticeable loss of detail. However, high-end cameras let you save all the details in a RAW format. With scanners you can select TIFF (TIF) for that. Another option is BMP with compression, which gives some file size reduction without any loss of detail.