Enhance Video/Audio Quality of Old Movies
There are some filters you can use, specifically speaking in regards to Premiere Pro editing, that can enhance old video/film content. These include applying some basic color correction, adjusting gamma, saturation, and using a 3 way color corrector or a filter plugin like Colorista to enhance the color balance per RGB channel. 

There is also "Magic Bullet's" Instant HD filter, which is part of their Shooter Suite. 

The filter does a slightly better job than simply scaling to frame size (dropping a small resolution video onto an HD sequence for example. It applies some sharpening, and contrast enhancements; however, the rendering time is intense. 

FOR THE BEST OUTPUT:

When I need to get the best output possible, this is the workflow I follow:


Using PPro or Adobe Media Encoder; export your video file as a JPG or preferably (if you have the HD space) TIF sequence. 
Go into Bridge, and open one of your JPG/TIF files using right click "Open in Camera Raw". Adjust your color, sharpening, clarity, vibrance, you can also, from with "Camera Raw", add "reduce noise" which can greatly help. 


The ability to enhance the frame from within Camera Raw is far more powerful than what you will get out of the box with Premiere. 


Then click Done to apply the "Develop Settings". 
Right Click - > Develop Settings -> Copy Develop Settings
Select all the frames (TIF/JPG) that are your exported sequence (there will be 1000s. 
Right Click -> Develop Settings -> Paste Develop Settings


NEXT


Use the Image Processor in Photoshop, it's under File -> Automate. 
Choose all the files you've applied your Develop Settings, and Check "Resize to 1920 H by 1920 W. 
Now run the Automation App. 


It might take a while, but you will end up with a folder filled with enlarged JPG/TIF files, with your develop settings applied. 


Lastly, go back into Premiere -> Import -> JPG File number 00001, and check the box at the bottom "Image Sequence". 


It will import all your frames as a video file. 

This is the best way to get the best results. But plan for overnight runs for the process of Automation of the Photoshop files from original to enlarged. 

Hope this helps.