feat: Migrate from NextAuth to Better Auth

Backend changes:
- Add better-auth and pg packages
- Create Better Auth instance with PostgreSQL adapter
- Add Better Auth route handler at /api/auth/*
- Create migration script for Better Auth database schema
- Update main index to use Better Auth routes instead of custom auth
- Configure email/password and OAuth (GitHub/Google) providers

Frontend changes:
- Add better-auth client
- Create Better Auth client instance configuration
- Update lib/auth.ts to use Better Auth session
- Rewrite sign-in page with Better Auth methods
- Rewrite sign-up page with Better Auth methods
- Remove NextAuth route handler

Documentation:
- Add comprehensive migration guide with setup instructions
- Document environment variables and API endpoints
- Include testing checklist and rollback plan

Benefits:
- Unified authentication for both Elysia backend and Next.js frontend
- Database-backed sessions (more secure than JWT)
- Better TypeScript support
- Extensible plugin system for future features
- Active development and frequent updates

frontend/lib/auth.ts

@@ -1,12 +1,12 @@
-import { auth } from '@/app/api/auth/[...nextauth]/route'
-import { redirect } from 'next/navigation'
+import { authClient } from '@/lib/better-auth';
+import { redirect } from 'next/navigation';
 
 export async function requireAuth() {
-  const session = await auth()
+  const { data: session } = await authClient.getSession();
 
   if (!session || !session.user) {
-    redirect('/auth/signin')
+    redirect('/auth/signin');
   }
 
-  return session
+  return session;
 }