Slow Toilet Tank Fill

One of our toilets' tanks takes forever to fill. And it seems to have a leak because it keeps on filling forever. Currently we have the water to it shut off and it is out of use. I want to fix it. I believe I need to change the entire system that's inside the tank. Can you tell I have never done any plumbing repair before? From the attached image can you tell me what do I need to replace? What tools I will need? Any other tips?

(I am on a DIY roll today. I have tasked myself to fix two other plumbing issues at our home. If you want to help, please click on my name and see my other two questions. Thanks. )

Thank you for your help.