The newest version you can run is 8.2, if you want to go to 8.3 or higher when they are released you will need to upgrade the RAM.
If you have an active smartnet agreement, then you can download the software from cisco.com. If not, then I am not sure.
upgrading the software is the easy part: upload the file to the flash, set the new boot image, remove the entry for the old boot image (can have more than one boot system command) and then reboot the device.
If you are using the ASDM:
Upload: Tools > File Management > File Transfer
Set the boot image: Configuration > Device Management > System Image/Configuration > Boot Image/Configuration
(same screen lets you set the ASDM image path. just upload the file, change the image path, close the ASDM and relaunch to use the new ASDM code, dont have to restart the device)
for the CLI: (use the file names on your system)
upload the file to the flash
enter config mode
boot system flash:/asa822-4-k8.bin
no boot system flash:/asa723-k8.bin
then restart when you are ready (wont change anything untill the device restarts)
upgrading the config, different story. read the release notes and make sure that your current config will be compatible with the new code. I went from a PIX right into an ASA with v8 software so I did a brand new config didn't have to worry about this part.
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Josh