I have downloaded some third party software, should I just put under
/Applications or should I put under my home folder, e.g.
It depends on what applications you want to keep to yourself and which ones you want to make accessible to all users:
I tried separating "personal" apps to ~/Applications/ and eventually gave it up after awhile because I found it to be annoying.
It turned out to be a lot hassle in little ways, and finally I just moved them all. I kept finding apps which only looked in /Applications/ so I had to keep tweaking settings.
More and more apps include checks to see if they are in /Applications/ and offer to move themselves, which meant even more annoyances.
If you install apps, such as Microsoft Office, which come with installers, you probably won't even have the choice.
But your experience won't necessarily be mine.
I'd put them into ~/Applications/ and then if you decide that you want to change it later, drag them all to /Applications/. It's easier to do that than try to separate them later because some apps do install themselves to /Applications/ and won't handle being moved somewhere else.
I just put everything in the application folder. I used to try to sort things but had issues with applications that do not like being an any folder other than applications (MS Office, Adobe programs, etc). Since I couldn't easily move these around, my organization scheme was incomplete and I just gave up.
This is probably sacrilege but you might want to try using Launchpad. The folders you create there aren't actual folders in the file system and therefore won't break picky applications that absolutely must live in applications. Mountain Lion also added a few new features to Launchpad that make it more useable, such as searching.