Here's the simplest way that I know of to speed up an iPhone (or for that matter, and iPad).
Wipe the device and reload your data from a backup.
Yes, it really is that simple.
I tend to do this every year for my devices, and not only do I find that I have more free space available (because app caches aren't part of the backup) but the system feels snappier and more responsive.
But before you do anything, first MAKE SURE YOU HAVE A BACKUP! That backup can either be up on iCloud or you can do one using iTunes. If you wipe your iPhone and you don't have a backup then you're going to have to reload your apps and information manually, and that's going to take you a lot longer than recovering from a backup.
The wipe process is easy to carry out:
1. Familiarize yourself with the process before you start.
2. MAKE SURE YOU HAVE A BACKUP!
3. Once that is done, wipe your device (Settings > General > Reset > Erase All Content and Settings, and from there you'll be guided through the wiping process).
4. Let the reset process do its thing. This shouldn't take long.
5. Now you're ready to recover your data from a backup.