Update usage of deprecated commands in GitHub actions

[aaemnnosttv]

Bug Description

See https://github.blog/changelog/2022-10-11-github-actions-deprecating-save-state-and-set-output-commands/

This also applies to dependencies that need upgrading in some cases.

For example, the GitHub actions/cache@v2 action raises these deprecation messages as well. See actions/cache#953

The fix for this only seems to exist in the newer v3 of the action, so we should upgrade to that.

These commands will stop working as of June 1st, 2023

Screenshots

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Acceptance criteria

• All usage of the deprecated save-state and set-output commands in GitHub actions should be updated to use non-deprecated methods (more information)
  • This includes upgrading any actions as needed

Implementation Brief

• Update the ".yml" files in .github/workflows to
  • use actions/cache@v3 instead of actions/cache@v2
  • replace set-output and save-state (if any) as per the docs.

Test Coverage

• N/A

QA Brief

QA:Eng

• Check the PR and make sure there's no deprecation warnings related to commands.
• Once merged, also check the tests on the merge commit to make sure everything is ok.

Changelog entry

• N/A.

[eugene-manuilov]

IB ✔️

[techanvil]

@kuasha420, apologies but I've merged this without spotting an indentation error, which is now preventing the workflow from running on develop, see https://github.com/google/site-kit-wp/pull/6190/files#r1029288649

Please could you file a followup PR to address this?

[techanvil]

QA ✅

• Confirmed that deprecated save-state and set-output warnings are no longer occurring since merging this to develop.
• For example here's the E2E (WordPress 5.2, AMP 1.x) workflow before and after:

Before:

After:

• Checked across all workflows.
• Note, we still have the "Node.js 12 actions are deprecated..." warning but this will be address in Update GitHub actions using deprecated runtimes #6188.