My computer often takes an unreasonable amount of time to connect to the wired gigabit etherenet network after booting the PC or after coming back from standby, usually if some time has passed since the computer has been powered off or set to standby. What logging or troubleshooting options are available to help find the source of this problem?
I would look at what you have built in to Windows first and to take note that a slow machine/connection may not be caused by an error per se and so logs may not be the answer.
Look at the event viewer.
When the machine boots up to Windows, open task manager and the performance tab - see how much CPU is being used. You can also use the Processes tab to see what programs are using most.
Type msconfig into the search box and see how many programs are running at start up (startup tab - but only edit if confident to do so).
Download a program like CCleaner and "clean" your machine (I won't give a tutorial on how to use it).