Where do you plan to come to Weligama from? If it's from Colombo, you surely have a lot of transportation modes. You can come to Weligama by train easily and the train ride will just be around 4.5 hours long. The second option is for those who won't prefer the train service in Sri Lanka. They have the option of catching a bus that leaves for Matara from the Fort bus station, and guess what... this bus ride can be quicker than the train ride. It will just take 3-4 hours for the whole journey, and if the bus you take is an A/C one that goes along the High Way, the ride will get much quicker.
Yes, you also have the option to take a taxi/cab from Colombo, but this option can be very expensive. When you come to Weligama, getting around the city is hassle-free as you have tuk-tuks for your assistance any time of the day.