Phones typically begin to slow down as their internal storage fills up. Try deleting unused apps and photos, or moving them to the cloud or a microSD card. In addition, you should close open apps that you are no longer using, delete app cache, and limit the use of live wallpapers.
An app’s cached data can be deleted by going to Settings > Apps, selecting a certain app, and choosing the Clear Cache option. Programs such as App Cache Cleaner and Clean Master, both of which are available for free from the Google Play store, can also be used to automate the process of clearing cache.
For more information, visit this article for speeding up your Android device.