In the city, the most economical and common mode is by 3 wheeler taxis (Tuck-Tuck as they are called). Be ready to hold on to the seat as the tuck-tuks zig-zag through the traffic like a hot knife though butter!
For longer distance (intercity), you can hire a car with driver for about 40$ a day, or you can take one of the trains in the vast train network in Sri Lanka. Or catch a local bus.
A MUST do in Sri Lanka is to take the train to the hills, it is considered one of the top most beautiful train rides in the world. You can visit here for train times https://eservices.railway.gov.lk/schedule/searchTrain.action?lang=en
4 words about hotels in Sri Lanka – Out of this world. There are many options depending on your budget. For us https://www.booking.com/ has been almost the only source for hotel bookings everywhere in the world except Cuba.