doc: Update README with new llms.txt approach, coding with AI guide, and other fixes

[koverholt]

Based on the updated approach to llms.txt in ADK docs in google/adk-docs#1023, this PR:

• Updates vibe coding section to use new llms.txt approach, and link to new "Coding with AI" guide in ADK docs, with steps for Gemini CLI and Antigravity
• Updates links in README to auto-generated llms.txt and llms-full.txt in the ADK docs site
• Removes manually generated llms.txt and llms-full.txt in this repo, since these are now autogenerated in ADK docs
• Various fixes / formatting updates in README

Checklist

• I have read the CONTRIBUTING.md document.
• I have performed a self-review of my own code.
• I have commented my code, particularly in hard-to-understand areas.
• I have added tests that prove my fix is effective or that my feature works.
• New and existing unit tests pass locally with my changes.
• I have manually tested my changes end-to-end.
• Any dependent changes have been merged and published in downstream modules.

[gemini-code-assist]

Summary of Changes

Hello @koverholt, I'm Gemini Code Assist¹! I'm currently reviewing this pull request and will post my feedback shortly. In the meantime, here's a summary to help you and other reviewers quickly get up to speed!

This pull request primarily updates the project's README.md to align with recent changes in the ADK documentation ecosystem. It modifies how AI-assisted coding context files (llms.txt and llms-full.txt) are referenced, moving from local copies to external, auto-generated versions. Additionally, it introduces a new guide for 'Coding with AI' and includes various textual and formatting improvements to the README.

Highlights

• Updated llms.txt approach: The README now reflects the new approach for llms.txt and llms-full.txt, which are now auto-generated and hosted in the ADK documentation repository.
• New 'Coding with AI' guide: A new section on 'Vibe Coding' has been added to the README, linking to a dedicated 'Coding with AI' guide in the ADK docs, which provides instructions for using these files with tools like Gemini CLI and Antigravity.
• Removed local llms.txt files: The manually generated llms.txt and llms-full.txt files have been removed from this repository, as they are now sourced from the ADK documentation site.
• README formatting and grammar fixes: Several minor fixes and formatting updates have been applied throughout the README to improve readability and accuracy.

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[gemini-code-assist]

Code Review

This pull request updates the README.md to reflect the new approach for llms.txt files, which are now auto-generated and hosted in the ADK docs. The local llms.txt and llms-full.txt files have been removed accordingly. The changes also include several wording and formatting improvements in the README. The updates are clear and improve the documentation. I have one minor suggestion to improve the phrasing in the 'Contributing' section for better readability.