Update Yosemite 10.10.5 to 10.13.6 or higher

https://support.apple.com/en-us/HT204915

Check out the lower part, I believe it allows you to set up your phone for receiving verification codes.

Upgrade to two-factor authentication on the web

1. Go to appleid.apple.com, then sign in with your Apple ID.
2. Answer your security questions, then tap Continue.
3. You'll see a prompt to upgrade your account security. Tap Continue.
4. Click Upgrade Account Security. 
5. Enter the phone number where you want to receive verification codes when you sign in. You can choose to receive the codes by text message or automated phone call.
6. Click Continue.
7. Enter the verification code to verify your phone number and turn on two-factor authentication.

From there on I believe you can upgrade your Mac to a later version of macOS. Big Sur should work on it.

Edit:

As VikingOSX already mentioned. Do keep in mind that from macOS Catalina onwards, that includes Big Sur, 32 bits software is no longer supported, and won't work. Mojave is the last version of macOS that fully supports 32 bits.

Another way of looking up which apps of yours are 32 bits is by opening up system information from apple icon in menu -> about this Mac -> system information.

Then scroll down to

software -> apps.

Made the screenshot on Big Sur, but should be similar on older versions. Older versions will show 32 or 64 bit apps in the type list (here Intel or Universal). 32 bit apps won't work on Catalina and later.

Why don't you land on Mojave until you sort out your 32/64-bit application standing? Use the Safari browser (only) to access How to get old versions of macOS, and its internal links to Mojave. You would need 45GB of available storage before you could upgrade to Big Sur, and none of your existing 32-bit applications would be supported. Consider running the free Go64 application, which can show you what 32-bit applications that you presently have.