Replacing the battery obviously comes with a few risks of losing data... but you may have too little storage, too many apps, etc? When you are at the home screen click the home button twice to get the small menu appearing at the bottom of apps that you have recently closed. Every app added uses minimum 5% battery if it is left there, which means your device will black out because of battery being used by the apps. To delete them from the menu, hold on any one of the apps in the small menu. They should vibrate and a small red cross should appear at the corner of each one. Delete them!
DO NOT WORRY! DELETING THESE FROM THE SMALL MENU DOES NOT DELETE THEM FROM YOUR DEVICE! THE ARE STILL AT THE HOME SCREEN!
Remember to clear this menu regularly!
If this does not seem to work, try syncing to a mac laptop but if that has no effect either, contact your internet provider or if you use wifi get it repaired. This should be free if your device still carries a guarantee.