I keep getting NET::ERR_CERT_DATE_INVALID.

You are running a very outdated OS that has not received security updates for years which means Safari is way out of date which is a bad thing since the Internet is constantly changing. Websites now require security certificates in order to safely connect and a security certificate recently expired. The only third party browsers still supported on macOS 10.11 are the Chromium based browsers such as Google Chrome or Vivaldi.

Your laptop can be upgraded to macOS 10.13, but even that has not received security updates for over a year now, but at least it can also run the current version of Firefox as well as the Chromium based browsers. Safari is too out of date to really be useful.

You can try the instructions in this forum post for upgrading the root certificate:

https://discussions.apple.com/thread/253247723?answerId=256074053022#256074053022

Another option would be to install Linux on the laptop, but this will require you to learn a new operating system. Plus Linux is not for everyone, but Linux does provide a great way of extending the useful life of older computers. I still have some 20 year old computers working great and connecting to the Internet & websites because of using Linux.

Once you decide macOS is no longer working for you on this laptop even after upgrading to macOS 10.13 High Sierra, then you can try install Linux Mint, or one of the Ubuntu flavors such as Kubuntu or Ubuntu-MATE. These particular Linux distributions listed should be easy to install on a Mac and should just work on an older computer with limited resources. Stay away from the other Ubuntu flavors since they tend to require a lot more resources or they are not as easy to use for new Linux users. You can test drive these Linux distributions by selecting the "Live" mode when booting the respective Linux USB installer, but keep in mind that they will run extremely slow when booted from a USB stick.

Before attempting to install Linux make sure to have a plan to reinstall macOS in case you don't like using Linux. I highly recommend creating and testing a bootable macOS 10.13 USB installer since Internet Recovery Mode doesn't always work as expected. If the laptop shipped with DVDs, then make sure the OSX 10.6.6 DVD installer boots. Also make sure to transfer all important files to external media which can be accessed by any computer/device and any OS since Linux cannot access a Time Machine backup. You should also have a good normal backup in case you ever decide to restore the laptop to macOS.

How to create a bootable installer for macOS - Apple Support

How to reinstall macOS - Apple Support

How to partition & format a drive for installing macOS 10.6 to 10.10:

https://www.owcdigital.com/assets/support/support-formatting-and-migration/Mac_Formatting_6-10.pdf