Someone PLEASE Help Me my NEW iPhone 5S Having Glitches and Im Crying Right Now

I just got this as a replacement phone for the previous iPhone 5S which I SHOULD have kept that one since I wasn't having problems much. First of all, Twitter push notifications weren't working then I fixed it by going to reset settings..NOW when I'm in 3G or 4G LTE mode, at times I'm NOT getting push notificatoins whenever someone sends me messages!! it used to send me messages and push notifications for my previous iPhones!! How can I fix this problem?? Does it sound like a software bug problem? if I update to the latest iOS version, should it work?? This happened when I first used this phone. Do you think it SHOULDNT be the device?? I am very NERVOUS because I don't want to get stuck to this 2 yr contract my mom yelled at me saying i cant do anymore replacements because it will cost extra which it did!! (shipping fee and also 30 dollars) . Also I already went to settings and EVERYTHING is set to ON and when I have received push notification, sometimes sound doesn't appear so it sounds like a glitch..is this the device that's causing this or the software?? when I'm not on Wifi mode, on KIK Messenger and Facebook push notications did NOT appear even someone sent me message. I'm extremely feeling like crap because my mom told me I can't replace it due to so many extra money via shipment etc.