What SharePoint 2007 features will enable me to:
- Assign and track tasks where each task will require a sign off before it is considered complete.
- Enter and Store daily status logs (i.e. each shift / person can details events).
The native task list will allow you track individual items. You can attach simple Workflows to manage what happens as tasks are updated. SharePoint will track the last person to edit an item, which can serve as your digital signature of the person completing the task. Additionally you can enable auditing on the list if you need more granular insight into how is reading / modifying the lists.
Shared Blog template would indeed fit your daily status log requirements. You can additionally look at the Wiki template, or an simple issue tracking list. The issue tracking list has the benefit that you can associate other list items that may be relevant to an entry made. It really depends on your requirements for displaying the data, archiving the data, and working with the data.
I would encourage you to pick up this great resource by Robert Bougue http://www.sharepointshepherd.com/default.aspx
If you would consider a 3rd Party SharePoint add-on, then I would suggest you take a look at DocRead for SharePoint. DocRead allows you to 'send and track' reading tasks straight from a SharePoint document library. Automated e-mails also get fired to remind employees to sign-off on a task. We also have another add-on module released in 2013 Q1 to allow you to quiz and survey employees before they sign-off.
To read more, take a look at DocRead.
Disclaimer : I work for the company that sells DocRead.