Kars city centre is compact and walkable — the main sights (castle, museum, cathedral, cheese shops) are all within 20 minutes on foot of each other. The Russian grid plan makes orientation easy.
For Ani, a taxi from the main square is the standard approach. Agree a price for a half-day round trip including waiting time at the ruins — expect to negotiate. Some guesthouses arrange shared taxis with other guests, which reduces the cost. There is limited dolmuş service toward Ani from the bus station, but it does not reliably serve the ruins directly.