Basically you can tell if AVG is working by double clicking the AVG tray icon and checking if all components are in "green" state.
As for the slowdown, please provide us with more information:
Windows version (2000, XP, Vista, Windows 7, etc...)
Hardware specifications (Amount of RAM and CPU speed)
AVG version and virus db version (found in the lower left hand corner of the AVG windows)
Other antivirus software installed / previously uninstalled (if any)
Other protection software installed
Is there any infection detected with AVG full computer scan?
It is also possible to slow down the computer when too many programs are constantly running in background. You can check what programs are being run at startup this way:
Type "msconfig" and press enter
Choose startup tab and uncheck any unneeded programs
Reboot your computer
I've used AVG for years, but how do I know it's working? Daily scans take forever now and computer is very sluggish during that time.