I expect you'll get many different suggestions/opinions on this. Here are some quick thoughts:
Many customers I work with use external monitoring tools or services that make requests to the LTM VIPs or GTM WIPs. Ideally they're checking dynamically generated content that touches the database to track client experiences. This is one important view of the entire health of the application stack including the F5's.
You can also monitor CPU/memory usage via SNMP or CLI tools. For spot checks from the CLI standard Linux tools help, along with tmstat.
Aaron
Cirrostratus