Sri Lanka is a small island and you can easily travel around it using public transport. It's not difficult to organise it yourself and to create your own itinerary.
However, you should note that transport times are not like those in the UK or Europe. For example, to cross the island from east to west by train, which is about 250 kilometres, takes 8 or 9 hours.
Whether you want to travel the coast, lose yourself in the mountains, or make lots of stop-overs, you will always find a means of transport.
There are few internal air connections in Sri Lanka. The army, and a subsidiary of Sri Lankan Airlines, use small planes to certain destinations (Jaffna, Galle, Trincomalee...) Last year an international airport opened at Mattara on the south coast.
With an international driving licence you can rent a car and travel around the country. However, in most cases travellers choose to rent a car with a driver because the prices are very accessible. It costs around 6,000 rupees a day. This solution is particularly interesting for families who are travelling or small groups. It's important to go to a local agency to get information first because not all drivers are the same!
The most enjoyable way to discover the countryside is to travel through Sri Lanka by train. The country has an excellent railway network covering most of it. There are three major lines: the Blue line, which serves the north and east of the island, the Green, which joins Colombo to the mountains, and the Red, which runs along the west coast from Periynagavitu to Mattara. The only downside is that the trains are slow and often late. On the other hand, the fares are very low, even in first class. You can find all the information you need on Sri Lanka Railways website.
There are also long distance buses and minibuses, both public and private, which go to nearly all Sri Lanka's towns. They are usually crowded and to be avoided if you are afraid of road travel. Traffic in Sri Lanka is chaotic and the drivers are paid by the number of journeys they complete per day. You usually buy your tickets from the driver on board.
Most of the country's towns are very small and can easily be discovered on foot or by bike. In the big conurbations there are several means of transport.
There are buses in all the towns and villages of the island. They are always crowded and it can be difficult to know which one to take because their destinations aren't always marked in English. Don't hesitate to ask passers by to help.
There are taxis in all the big towns. They don't all have meters so you should allow about 100 rupees per kilometre..
Tuk-tuks or bajaj are everywhere, in towns, by the sea, in the countryside. It's one of the most common means of transport on the island. You have to negotiate very firmly with the driver who won't hesitate to increase the price if he only goes 50 metres further than agreed. In Colombo nowadays there are tuk-tuks with meters.