Step 1: Historical first implementation.
Panchayati Raj was inaugurated in Rajasthan on 2 October 1959 at Nagaur district—considered the first implementation in India.
Step 2: Context and follow-up.
Following the recommendations of the Balwantrai Mehta Committee (1957), states began adopting the three-tier system; Andhra Pradesh followed soon after Rajasthan.
Step 3: Eliminate distractors.
Punjab, Bihar, and Gujarat were not the first to implement the system.
Step 4: Conclude.
\[
\boxed{\text{Rajasthan (Option C) — first state to start Panchayati Raj (Nagaur, 1959).}}
\]