Any time someone does an install of a newer OS and the system slows down, it's usually something like a bad kext file that was migrated and is now incompatible, excessive MDS indexing as the new OS indexes everything, the way the user is using the system, or something (like anti virus software) that's effectively blocking applications or system processes. It's possible the drive could be going bad, but the odds of it going bad right when an OS upgrade occurred isn't that likely - possible, but not likely.
You might want to visit this site:
http://scsc-online.com/How-To.html
Scroll down to the article titled
The Signs and Symptoms of Hard Disk and Solid State Drive Problems, and click on the link. Before assuming it's HDD, SSD, or even a system problem I would read the intro page then read the links associated with
User Problems and
Software Problems.
If you're using a beta version of the OS then it will gather information from your system which can take a lot of time and bog the system down. I wouldn't recommend using mdutil to enable/disable indexing because it will only do so on user drives and will allow indexing on Time Machine drives to continue. If mdutil was used to disable indexing, when enabled it will start re-indexing everything from scratch and bottleneck the system. One guy somewhere on this site said it took an earlier versions of OS X something like 2 or 3 days for his system to finish it's initial indexing because he had lots of data.
Hope this helps.