Can't sign in to iTunes&App Store on iPad Air

I bought my iPad Air a little over a year ago and I've never been able to sign in to iTunes & App Store. I have to purchase, download and update apps through my iPhone or computer. App Store and iTunes are the only things I have trouble with. It's extremely frustrating and makes me feel like I truly wasted my money. I've seen people post similar issues but none of the solutions that have been suggested to them have worked for me. When I try to sign in to iTunes and App Store in Settings, I put in my password but nothing happens. The "sign in" button goes gray for a moment but then it goes blue again and I'm still not signed in. Connecting my iPad Air to my laptop just to update apps and also to download new apps pretty much defeats the purpose of having an iPad Air. Does anybody at Apple have a solution for this problem? Please help if you can