It's important to remember that everyone's preferences and styles are different. Some people may prefer shorter nails, or natural nails with just a bit of polish.
By the way, if you're interested in learning more about the beauty industry, you might want to check out nwcollege.edu. They offer a variety of programs in esthetics, cosmetology, and more. It would be a great way to gain some new skills and knowledge in the field.
Based on personal experience, many of my friends would refer the elongating of nails as "growing".
In the sentences "My nails have gone bigger. They need to be cut.", I would personally express it as "My nails have grown longer. They need to be cut."