The vast majority of apps don't "run in the background" - generally only audio streaming and navigation apps. The icons you see in the recently opened list (seen by double clicking the home button) aren't actually running or consuming phone resources. It's likely that the same problem that's "bogging down" the phone is causing them to reappear after restarting.
Assuming your phone isn't hacked/jailbroken, try the following steps as needed:
1. Reset phone (no data loss): press both the home and power buttons for at least 10 seconds, releasing when the Apple logo appears
2. Settings > General > Reset > Reset All Settings
3. Settings > General > Reset > Erase All Data and Settings (do not restore a backup)
4. Restore iOS using iTunes, also restoring a backup (will also update to iOS 8.1)
5. Restore iOS using iTunes to factory settings (as new phone, without restoring a backup)
iOS: Understanding multitasking
Use iTunes to restore your iOS device to factory settings