iOS 7 should make iPhone 4 run the best it can. Part of 7’s development was about wringing the last bit of performance out of relic iPhone 4 and 4S. If you haven’t installed it do so.
iOS 8 will likely not be recommended for iPhone 4s.
Removing background programs from the multitasking tray will not improve performance in the long ish term.
Unless an app specifically operates in the background (something you can control in Settings) an app will go dormant by about 5 minutes time. Restarting the application from scratch will draw on the battery more than if it is awakened from a dormant state. Clearing the multitasking tray is usually unnecessary.
Apple is obsessed with setting most things for maximum battery life and stability. The typical user doesn’t need to maintain anything. Configuring Settings for maximum battery life excepted.
Rebooting the phone will usually clear any minor issues and should be done every several days. It clears some unneeded files and makes the phone scan for changes in the cell network.
Hard resets should only be done if rebooting fails to fix a problem. Routine hard resets are not recommended. You can often save yourself time by hard resetting before calling support as very often that will be support’s first line of action. Doing so will not cause you to lose any data or settings.
I’ve seen the word “restore” used a few times in this thread. Restore means to remove software and firmware from the phone and reinstall. This should only be done to address major problems. It will require that you back up all applications and data first, then reload.
Some applications are notorious battery drains. Facebook is the best example. Facebook wants to be the only app. It gets insecure so quickly. “Why aren’t you using me constantly?” it whimpers. It is the AOL of iOS applications. Do yourself a favor and remove it. Access Facebook through a browser instead. I won’t even get into the security matters of Facebook and Facebook Messenger.