Add structured logging and centralized error handling

- Initialize slog in the serve command with terminal/file support
- Introduce `AppError` with HTTP status for unified service-layer errors
- Replace ad-hoc `api.Error` calls with `api.RespondError`
- Wrap internal errors with `model.NewInternalError` and add reference
  IDs
- Add `RequestID` middleware and switch router from `gin.Default` to
  `gin.New`

internal/model/errors.go

@@ -1,6 +1,10 @@
 package model
 
-import "errors"
+import (
+	"errors"
+	"fmt"
+	"net/http"
+)
 
 var (
 	ErrNotFound     = errors.New("resource not found")
@@ -8,3 +12,25 @@ var (
 	ErrUnauthorized = errors.New("unauthorized")
 	ErrForbidden    = errors.New("forbidden")
 )
+
+// AppError is a service-layer error that carries an HTTP status, a user-safe
+// message, and an optional internal cause for logging. Handlers unpack it
+// transparently via api.RespondError.
+type AppError struct {
+	Status  int    // HTTP status code
+	Message string // safe for API response
+	Err     error  // internal cause (nil = user-caused, skip logging)
+}
+
+func (e *AppError) Error() string { return e.Message }
+func (e *AppError) Unwrap() error { return e.Err }
+
+// NewInternalError creates a 500 AppError with a wrapped internal cause.
+// msg describes the operation that failed; cause is the underlying error.
+func NewInternalError(msg string, cause error) *AppError {
+	return &AppError{
+		Status:  http.StatusInternalServerError,
+		Message: "internal server error",
+		Err:     fmt.Errorf("%s: %w", msg, cause),
+	}
+}