IMac w Mac Os 10.5.8 needs to upgrade. HOW?

Question

iMac w Mac Os 10.5.8 needs to upgrade. HOW?
My Time Warner account for email just stopped allowing my MACMAIL program to send mail out. We tried to change settings but that just stopped
email in AND out. Put it back to the original settings. No answer as to why. I assume the OS system is too archaic, and will soon be bounced out.
How do I upgrade?  Never have before. My old PowerMac is still OS9, and has thousands of dollars of working software with NO EMAILS
or INTERNET available. Chose to leave System 9.  Can I upgrade from 10.5.8 to the newer ones, and HOW????  HELP PLEASE!
Thanks!

Answer 1

Upgrading to Snow Leopard
You can purchase Snow Leopard through the Apple Store: Mac OS X 10.6 Snow Leopard - Apple Store (U.S.). The price is $19.99 plus tax. You will be sent physical media by mail after placing your order.
After you install Snow Leopard you will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store. Access to the App Store enables you to download Mavericks if your computer meets the requirements.
     Snow Leopard General Requirements
       1. Mac computer with an Intel processor
       2. 1GB of memory
       3. 5GB of available disk space
       4. DVD drive for installation
       5. Some features require a compatible Internet service provider;
           fees may apply.
       6. Some features require Apple’s iCloud services; fees and
           terms apply.
Upgrading to Lion
If your computer does not meet the requirements to install Mavericks, it may still meet the requirements to install Lion.
You can purchase Lion at the Online Apple Store. The cost is $19.99 (as it was before) plus tax.  It's a download. You will get an email containing a redemption code that you then use at the Mac App Store to download Lion. Save a copy of that installer to your Downloads folder because the installer deletes itself at the end of the installation.
     Lion System Requirements
       1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7,
           or Xeon processor
       2. 2GB of memory
       3. OS X v10.6.6 or later (v10.6.8 recommended)
       4. 7GB of available space
       5. Some features require an Apple ID; terms apply.
Upgrading to Yosemite
You can upgrade to Yosemite from Lion or directly from Snow Leopard. Yosemite can be downloaded from the Mac App Store for FREE.
Upgrading to Yosemite
To upgrade to Yosemite you must have Snow Leopard 10.6.8 or Lion installed. Download Yosemite from the App Store. Sign in using your Apple ID. Yosemite is free. The file is quite large, over 5 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.
    OS X Mavericks/Yosemite - System Requirements
      Macs that can be upgraded to OS X Yosemite
         1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later
         2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later
         3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later
         4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later
         5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later
         6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later
         7. Xserve (Early 2009) - Model Identifier 3,1 or later
To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.
     Are my applications compatible?
         See App Compatibility Table - RoaringApps.