I have a site on Magento 126.96.36.199. I recently got a notification to upgrade to 1.9.3 and SUPEE-8788. I am used to Wordpress where you get one update and install it and that is it. Do I need to install the Supee patch if I install 1.9.3? Are they separate? Should I install one before the other?
You only need to add SUPEE-8788 patch if you're staying with 188.8.131.52. If upgrading to 1.9.3 the upgraded version includes security patch. Upgrading Magento isn't simple as wordpress unfortunately and in some cases it can cause few things misbehaving here and there which once upgraded can be fixed one by one with community's help (thanks to StackExchange). In your case though, you're not running too old Magento so risk is low yet you should be making backups of both files and database before upgrading. Safe option is to create a local installation from production files and db, upgrade to latest Magento, test, fix(if anything breaks like forms etc.) and when happy upload it to production.