Can you receive traffic advisories without a specific destination or intention? I’m wondering if it is possible to call up the local approach facility and ask them for flight following even if you are just doing sightseeing or something. Do you need to tell them your intended route, altitude, and destination? How would you ask them for flight following without a specific destination?
You call up ATC, tell them that you are out sight-seeing and would like flight following and traffic advisories.  On a workload permitting basis ATC would provide the service.
Here is an excerpt from the FAA Air Traffic Control Handbook:

Section 6. Basic Radar Service to VFR Aircraft- Terminal
7-6-1. APPLICATION
a. Basic radar services for VFR aircraft must include:

Safety alerts.
Traffic advisories.
(...)


This is a very common request and depending on the location there should be very little difficulty in receiving this service.