I noticed on another android phone there were "updates" such as new Youtube, Gmail, etc. I never received a notification for these on my device. Are these delivered via Google Market, or is there another app I can use to update these other than Google Market?
App updates notification can come from the app themselves, but most commonly they come from the Play Store. Play Store will notify you for any updates to system apps (i.e. apps that originally comes with the phone) and apps that are installed through the Play Store; it won't notify you about updates to side-loaded apps our apps installed by third party market.
Third party market may also have their own update notification mechanism; which is completely unrelated to the Play Store update mechanism. They may or may not provide update to system apps, although most don't. Amazon App Market for example also provide updates notification. Note that update notification on third party market depends on the app developer updating the app on the third party market, some developers might update their app on Play but not on third party market or vice versa for various reasons.
I noticed on another android phone there were "updates" such as new Youtube, Gmail, etc. I never received a notification for these on my device.
Go to Play Store App in your device and hit the Menu button and Go to My Apps you can see the installed apps list here. select any app and you can see the Check box
Allow Automatic updating below the application screen shots. Check this and you will be notify every time when the updates are available and it will automatically update if you are connected with any Network (EDGE/Wi-Fi/3G).
Are these delivered via Google Market.
Ofcourse, they are delivered via Google Market.
or is there another app I can use to update these other than Google Market?
No, There are no other apps that you can use to update these apps. if there will be any app that offers updates they will ultimately redirected to the Google Market.