How can I figure out what my password is and keep it unlocked?  Every time I do a system update, it requires the password and sends me into cyber trouble with the remember my keychain access, requiring it every time I log in.  A real hassle to b

How can I find out what my password on the computer is?  I had to change it due to my teenagers helping themselves, and now every time there is a system update, I have to have the password, which I cannot remember or find.  It then wants it every time I log in and pops up Key Chain Access as well.  Very annoying.  I do not have the disc to reload it and fear I would loose everything on my computer as there is also trouble with the back up system.  Too many issues and not enough solutions.  Many thanks in advance!

I do not have the disc to reload
Why not?  You need your system dvds to troubleshoot & to reset/change passwords in view of your current OS listed in your  profile. 
You can get replacement System Install & Restore CD/DVDs from Apple's Customer Support - in the US, (800) 767-2775 - for a nominal S&H fee. You'll need to have the model and/or serial number of your Mac available.
If you're not in the US, you may need to go through the regional Apple Store that serves your location to find the contact number. Here's a list of links to all of those - Another resource:  International Support Phone #s.
I have to have the password, which I cannot remember or find.
When selecting passwords, make sure it's one that you will NEVER forget AND no one else can figure out. 
Old school--- > Print it out & keep in a safe place.  A place that ONLY you know about AND never forget.
New school---> Get a password manager utility.  Highly recommend 1Password which is shareware.  Do a Google search for free password managers.