[Search] Refactor get started

[leemthompo]

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Key changes:

• Added new /solutions/search/api-quickstarts.md as a central hub for API tutorials
• Moved core tutorials into /solutions/search/ directory:
  • elasticsearch-basics-quickstart.md (was getting-started.md)
  • querydsl-full-text-filter-tutorial.md
  • run-elasticsearch-locally.md
• Consolidated connection info into single search-connection-details.md
• Moved serverless content to serverless-elasticsearch-get-started.md
• Updated toc.yml paths and structure to match new organization
• Improved navigation structure and reduced redundancy in getting started content
• Cleaned up filenames/titles/etc. generally where possible

Followup work

• Review search/get-started section for typos/formatting errors etc.
• Verify all images
• Write get started overview blurb