To get to Carmarthen, you can take the A40 road if you're driving from the east or west. If you're coming from the north, you can take the A485 or the A485/A484. From the south, take the A484.
If you're taking the train, you can get off at Carmarthen station, which is served by Transport for Wales. There are regular services to and from Swansea, Cardiff, and other towns in the area.
If you're coming from Swansea, you can take the A483, which takes about 30 minutes. If you're coming from Cardiff, take the A48 and the M4, which takes about 1 hour and 20 minutes. From Aberystwyth, take the A485, which takes about 1 hour and 20 minutes.
The closest airport to Carmarthen is Cardiff Airport, which is approximately 70 miles away. If you are driving, the most direct route would be to take the M4 west to Carmarthen. Alternatively, you can take a train from Cardiff Central Station to Carmarthen Station, which takes approximately 2 hours.