how to check hard drive health

This is unlikely caused by a bad hard drive.

Open Disk Utility. Select the topmost drive entry on the left sidebar. Check the status area at the bottom of the window. If SMART is "Verified," then your drive is OK.

If you are running OS X "Lion" or later, boot OS X Recovery by holding ⌘ and r (two fingers) while you start your Mac Pro. For Snow Leopard or earlier, start your Mac while holding the c key with its original OS X Install DVD inserted in its optical drive.

At the Mac OS X Utilities screen, select Disk Utility. Select your startup volume (usually named "Macintosh HD") and click the Repair Disk button. Describe any errors it reports in red. If Disk Utility reports "The volume Macintosh HD appears to be OK" in green then you can be reasonably (though not completely) assured your hard disk is in good working order.

- The SMART Status gives you some assurance that there are not massive numbers of Bad Blocks on the drive.

- The Disk Utility (Repair Disk) tells you the Directory [Scoreboard of what file is where] on the drive is in pretty good shape.

+ The two of them, used together, give you the warm fuzzy feeling that things are OK, without taking Hours to do so.

Normally if you have a hard drive problem it manifests itself with delays, periodic locks-ups, I/O errors, etc.during notrmal use, not just start ups and slow downs.

Never the less, you asked how you check the health of a drive, so I'll answer.

You can check the SMART status of your drive with Disk Utility or a free tool like smartctl in the smartmontools package. Disk Utility will typically report the SMART status of your drive as "verified" or a message associated with a type of failure right in the Disk Utility window. SMART status will be reported on most internal drives and Thunderbolt drives. More elaborate SMART information can often be obtained with newer OS versions by clicking on the "info" button. SMART is a reporting technology and will only report on errors after they occur. Many in the industry are highly critical of SMART, with a recent study by Google on hard drive problems indicating that it did a poor job indicating impending problems.

Other tools such as Scannerz (http://scsc-online.com/Scannerz.html), TechTool Pro (http://www.micromat.com/products/techtool-pro) and Drive Genius (http://www.prosofteng.com/products/drive_genius.php) do, IMHO, a better job of testing because surface scans can detect errors not detected by SMART yet, and all three of them do those tests. Personally I like Scannerz myself because it's to the point, hardware testing oriented, and can expose other system problems like cable problems, whereas the others are more like swiss army knife tools with a lot of features that may or may not be of use to end users. Some of the people on this site will swear by these tools, whereas others will swear at them. All three of these also monitor SMART status in one way or another.

With software tools out of the way, I'd have to agree with all previous posts, because it doesn't sound like it's probably a hard drive problem. Yes, it possibly could be, it's just that slow start ups and shutdowns are usually caused by too many apps being saved in an "on" state. Any time the system starts, it has to re-load them and resume them as best as it can, and then any time it gets shut down it must save all that information and this all takes time. (I'm assuming Lion or newer operating system is in use). This is often the most likely cause in Lion, Mountain Lion, and Mavericks.

There are other possibilities, but you should probably ellaborate on the problem a bit more.