Is there a best-practice method to work through a checklist?
The current FAA Advisory Circular dealing with SOPs in general, but only lightly touching upon checklists, is AC 120-71a. It is primarily aimed at Part 119 certificate holders who conduct operations under part 121, but is also recommended for operations under Parts 91, 125, and 135. It also changes the wording from Pilot Not Flying (PNF) to Pilot Monitoring (PM) in order to reinforce the idea that he or she is not just a passenger with a better view, but an integral part of the flight crew.

With respect to actual checklist usage, to the best that I've been able to find, the FAA does not issue a hard recommendation (air carrier checklists and procedures have to be approved by the FAA prior to certificate issuance anyways), but it does identify two primary methods of checklist usage: Challenge-Do-Verify (often used for abnormal and certain emergency procedures) and Do-Verify (often used for normal and time-critical emergency procedures), which relies on memorized flows followed by verification with the use of a checklist.

A 1995 Study issued by the FAA weighs in on the pros and cons of the two methods and surmises that, when used properly, the CDV method is less likely to result in missed items, but does allow for the fact that, in operational usage, pilots may batch items together, thus defeating the purpose of the systematic, step-by-step approach of the method.  

Stepping aside from the multi-crew environment, where having someone with the checklist in their hand and reading things off one by one can work quite well, my personal observation has been that, in a single-crew environment, pilots who rely on a CDV-like method (sometimes known as Read-Do) are more likely to end up falling behind in configuring their aircraft than those who use DV-like methods, assuming of course that the pilots actually know their flows and perform the verification properly (as in actually verify that everything is as it should be rather merely reading items off the list and trying to remember if they've accomplished them).

Lastly, with respect to the 'checked' vs. verbal confirmation of status, this is usually dependent on how critical and laborious the status confirmation is. An example would be with altitude setting verification, where it is not uncommon for the procedure to call for the PF to set the new altitude in the Mode Control Panel and, while pointing to it, call out the new altitude, which is then verbally confirmed by the PM. Every operator's experience and demands will be different here, but a number of best practices are recognized and the folks at the FAA, NBAA, FSF, AOPA, and many others are doing their best to publicize and promote them.

See here for the FAA's Industry Best Practices Reference List.

The checklist you've provided appears to be intended for multicrew operations. In this particular case, what would happen if the CDV method was to be used is that the Pilot Monitoring would read the items off one by one and the Pilot Flying would execute accordingly, verbally acknowledge doing so, the PM would quickly verify compliance and move on to the next line item.

In a DV environment, the Pilot Flying would go through the flow, and call for the checklist, at which point the Pilot Monitoring would begin to read off the items, and the Pilot Flying would verify that they have been complied with.

The line items that have BOTH in them are meant to be a special emphasis case in that both crew members are required to verify that the line item has been complied with, regardless of method used.