I am getting messages that my browser is outdated and needs to be updated. I go into System Preferences in the Apple Menu but I do not have the Software Update option. I don't know how to update the version of Safari (I have 10.1.2)on my laptop.

Check the computer requirements shown in each link to see if your computer is capable of running that OS version.

macOS Big Sur is compatible with these computers

Catalina

How to upgrade to macOS Mojave - Apple Support

How to get macOS High Sierra   

Sierra download

Larry2960 Said:

"I am getting messages that my browser is outdated and needs to be updated. I go into System Preferences in the Apple Menu but I do not have the Software Update option. I don't know how to update the version of Safari (I have 10.1.2)on my laptop. How do I update my software?"

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Time to Upgrade?

You macOS is obsolete (perhaps the Mac too?) That's a pretty old macOS. So, see if you can upgrade past macOS 10.10. Where to start: What is the model of your Mac? Please check, so we can see if it can be upgraded.

To Find Model Number:

1. Go to: Apple Menu
2. Select: About this Mac
3. View: second line-down

You are years out of date and upgrades are free.

Open the App Store and see what upgrades (not updates) will work with your machine.