There are quite a few options available for getting to the North East of Sri Lanka and Trincomalee depending on where you are in the country. Trincomalee is connected to the rest of the country by road, rail and airport so you have choices.
We were traveling to Trincomalee with quite a bit of luggage as we had our scuba diving equipment with us in addition to our personal travel packs. Also, coming from the south we decided that our best option is to head to Trincomalee from Colombo as options there will be best. We did find options ranging from a night train to busses and private cars, but eventually we decided to pick the private car as the most suitable option for the relatively long journey considering all the luggage we were carrying.
If you are taking public transport, be sure to reserve enough time, book in advance if possible and have patience – the journey is quite long.
Trains and busses are quite cheap options considering the distance. For a private car, you will however need to pay quite a bit more. Services are widely available and you will be able to negotiate a little from the original offers but expect to pay approximately 80-100 euro for the trip / car. You should be able to fit 3-4 people in a car and 5-7 people in a van, so eventually the per person price will be quite affordable.
For recommendations, ask your hotel or guest house for contacts as this way you are likely to get the best price. We also found affordable options online and it is worthwhile calling a few companies to compare quotes as the first company we called quoted a double price to the rest.