How to hook into "visiting the website"?
Using Rules Event "Drupal is initializing" (within "System") for sure will do the job:


It doesn't depend on login (can be used for any role).
It applies to any path (a node, a view, etc).


Should you want to do so, you can also add any desired Rules Conditions (eg to only consider selected paths, eventually using RegEx expression also).

Make it right before you make it better

I'm not sure this is the only way your question can be answered, but I do know that this Rules Event (Drupal is initializing) will do what you're asking for, which is to hook into "visiting the website" (i.e. visiting any page) (as in your question).

But, using this Rules Event also comes with a price: it will case this rule to be triggered on every page request (which is a pretty similar to your visiting any page). However, if this price is too high, maybe you want to consider any (or all?) of these compromises:


Add Rules Conditions such as "only for users with selected roles" (eg: authenticated, so that for Anonymous users the Rules Actions are not triggered).
Instead of any page, you only take into consideration something like "viewing content" (for which you know there are less expensive Rules Events available).