The easiest way is to have a taxi waiting for you, all the resort have this service and you will be paying from 50-60 € per taxi to go to Kalpitiya (around 3h30min). And 100 € to go to Talaimannar (8h). If you arrive during light time and you are not in a hurry you could find better deals as soon as you go to the taxi area outside of the airport. You will be paying around 40 € to Kalpitiya and 70€ to Mannar.
If you have time and you want your 1st Sri Lanka experience you can take the bus (it would cost around 8 €/pers to Kalpitiya, 12 €/pers to Talaimmanar). The buses do not have AC. Get out of the airport and walk straight for 5min to the main road (be ready to be told that you can not do that, and there are no buses to go to Kalpitya/Mannar…which is not true).
I reccommend you to have a notebook with everything written in a peper to show to them. Most of the bust drivers do not speak English. They will help you with the luggage, if there is space in the back of the bus they will open for you.
1st) Take the bus that goes to Negombo and stop at Kutanayake Junction (30ruppies). (It’s 10min drive) You can also do it with Tuck Tuck (300Rupies)
2nd) Take the bus that goes to Puttalam and stop at Palavi Junction (200 ruppies, 3h). They pass every 30min.
3rd) From Palavi Junction take the bus to Kalpitiya and you have to get out at Kuriyampity Junction ( 70 rupies, 1h). And then take a tuck to your resort.(300 rupies). If your resort is not close from this Junction carry on with the bus and jump out on the junction that is closes to your resort. Otherwhise you alway can get a tuck tuck from Kalpitiya to go to the resort.
1st) Same as for Kalpitiya.
2nd) Take the bus that goes to Puttalam (200 ruppies, 3h). They pass every 30min.
3rd) Get out of at the last stop: Puttalam. And take the bus to Mannar (250 ruppies, 4h).
4th) In Mannar change bus to Talaimmanar (100 rupies, 1h)
5th) In Talaimannar take a tuck tuck to your resort. So far the only kitesurfing resort is Vayu.