It's too hard to judge, an upgrade could simply be from 1 shared hosting provider to another, it's a tough comparison. Generally though when either you think you've outgrown it or are reaching limits, or when the provider thinks you've outgrown it (question this though as an upgrade may not even be needed, could just be upsell).
However, a VPS isn't really an upgrade from standard web hosting - it's not the same product.
A dedicated can probably be seen as an upgrade from a VPS though.
One common misconception is that you have to move through that cycle from shared to VPS to dedicated, when in reality a good shared platform could be all that's needed. Especially for a single website (dependent on traffic/purpose).
General rules doesn't work here - its as case-specific as it can get
It matters what script you are using to make your website, what number of visitors you are getting, connections, traffic stats, CPU used etc. All those influence the allocated resource on your hosting account. If you are using a control panel, like cPanel for example, you will be able to see those stats inside and you can monitor them frequently. As soon as you see that you are reaching your limits, its time to think about upgrading. Event though I also don't see the VPS as the natural upgrade progression from Shared, it is indeed something like a middle step between Shared and Dedicated, especially price-wise.
All in all just make sure you are aware of all limitations, resource-wise, you have with your hosting and make sure to upgrade before you can outgrow them and fall into suspensions, downtimes and other unpleasant occurances