The two most important tools to sharpen are scissors and rotary cutters. Remember, we already sharpened our needle by changing it!
For some reason, many of us tend to fight with our rotary cutters by making multiple passes over fabric or snipping a few spots that did not cut right. Or we press down harder as if more pressure will make it cut better. I know. I have done it. Those are signs that our rotary cutters need our help. They are screaming for a blade change. So rather than fight it. Give in and replace it. I am sure your fabric, your mat, your rotary cutter, and your arm will all thank you.
Rotary cutter blades are not to super expensive, but watch for specials, particularly at trade shows. Sometimes shops will have specials too. These are things I watch for and buy for later. I always need new blades.
Changing rotary cutter blades is not difficult, but it does require you to pay attention to avoid cutting yourself. As a picture is worth a thousand words, which is why I produced a short video about sharpening or changing your blades. The video also gives a little explanation on when to use the different cutters.
Sharpening scissors is a very different story. Given scissors can be expensive, we do not throw them out when they get dull. Usually, we take them to a person who specializes in sharpening. These people will visit guilds and shops for sharpening days. There are also ways to maintain your scissors at home. However, if they are seriously damaged (twisted) or serrated, you will want to see a professional. FabricFit has a nice tutorial with a few different sharpening methods, including when to use each method. They include video from a Bridal Sewing Techniques that was very helpful if you have an electronic sharpener.
Keep your tools sharp!