Improve/docs

[jhnwu3]

This pull request updates the documentation to make the PyHealth Research Initiative more accessible and engaging for prospective participants. The changes reorganize the quick navigation, clarify the program's mission and goals, provide detailed information about application steps and program structure, highlight recent achievements, and improve communication channels.

Navigation and Structure Improvements:

• The docs/index.rst quick navigation was reorganized for clarity: sections are now bolded, installation instructions are moved up, and the research initiative is more prominently featured.

Program Mission, Goals, and Achievements:

• Added a new "Mission & Goals" section to docs/research_initiative.rst, outlining the initiative’s purpose, focus on reproducible research, real-world impact, and collaboration with healthcare professionals.
• Highlighted recent successes, including publication of three papers at ML4H 2025, and clarified ongoing research cycles and targeted conferences.

Application Process and Participant Benefits:

• Replaced the old "Getting Involved" section with a detailed, step-by-step "How to Apply" guide, including Discord joining, application form submission, and PR requirements. Added links to bounty lists and contribution guides.
• Expanded the "What You'll Gain" section to enumerate tangible benefits: publication, hands-on experience, mentorship, collaborations, and impact.

Communication and Community:

• The "Contact" section was renamed "Stay Connected" and enhanced with clearer instructions, links to GitHub, mailing list subscription, and Discord for community support.