The first set of tools will not be that much of a surprise to most Windows admins, Sysinternals (Mark Russinovich, you're my hero). These tools are so awesome that Microsoft bought the company and Mark is now a technical fellow there. There are about 65 different utilities, some are great (process explorer) and some aren't (RegJump). The main ones I use are:
Process Explorer
This one is my favorite. Think of it as Task Manager on steroids. It will show everything you could ever want to know about a process including all handles and dlls that are in use.
Process Monitor
Shows real-time FS, Registry, TCP/UDP, and Thread/Process for all processes. This is a must if you ever want to know what a process is actually doing.
AutoRuns
Shows you what programs are configured to run during system bootup or login, and shows you the entries in the order Windows processes them. I have found some many compromised servers using this tool.
Strings
This is used to display all strings contained within a file. This is also handy for compromised systems.
As I said there are over 60 tools and these are just the tip of the iceberg, so check them out.
Another tool set I use are from NirSoft. These guys have some great tools that I can't ever seem to replace.
Currports
This is network monitoring software that displays the list of all currently opened TCP/IP and UDP ports on your local computer and the executable listening.
OpenedFilesView
Displays the list of all opened files on your system. For each opened file, additional information is displayed: handle value, read/write/delete access, file position, the process that opened the file. You can also close one or more opened files, or close the process that opened these files.
RegDllView
This tools shows all the object that are registered on the system. Is a must for anyone responsible for a web environment
NirSoft also has a full line for password recovery utilities that have saved me more than once.
As for built-in commands I use the following everyday (i'll let you Google them):
netstat
tasklist
taskkill
findstr
netsh
sc
net
I hope these tools can help you find the cause of any issues that you are trying to resolve as they have helped me for years.
Jeff