Spotlight is always the place to start. It's fast and frequently finds what you're looking for right off the bat.
Using Spotlight keywords to narrow the search may make it faster and will improve the accuracy of the search.
Even so Spotlight won't always produce the desired result. For instance system files are not automatically included in Spotlight searches.
Someone mentioned the Finder search. It should be noted that it is just another UI for Spotlight.
It's a good idea to have a means to search your hard drive(s) that does not depend upon Spotlight.
find utility is very useful, but it has its limitations and isn't all that fast.
Probably the best free find-utility for the Mac is EasyFind from DEVONTechnologies. You can download it directly or get it from the App-Store.
This utility does not depend upon Spotlight or any index when searching. It crawls the disk looking for hits and is faster than Unix's
To my recommendations I would add Find Any File. At $6.00 U.S. it's a bargain – it's faster than EasyFind – and with a password it will search as root.
Finally there is File Buddy – the venerable 900lb gorilla of Mac search utilities. At $39.95 U.S. it's anything but cheap, but it has many features that have kept me upgrading since the early 1990s.