There could be a few things that could be causing your issues
1) check for storage available on your phone? If your storage is full or over full it could cause all sorts of problems, its most common on the 8GB and 16GB units.
2) Check your iOS software version if it is out of date? If this is out of date then I suggest doing a Backup of the phone via iTunes ( make sure its up to date also) and then restore and update the phone. After done restore your iTunes backup to your phone.
(You can check both your iOS and storage by following : Settings > General > About. If you scroll down you will find all of the info for your phone.