The first and most popular way of celebrating Songkran is by throwing water.

Here is a snippet:

Songkran is called the water festival, or the water throwing festival, as throwing water is the major feature of the event. When considering the spiritual and religious aspects of the festival, some people may think it is rude and outrageous that people nowadays use buckets, garden hoses, water balloons and water pistols to drench each other. But the throwing of water has now become the most fun part of the festival during this hottest month of the year - around 40 C, or 100 F. Young people typically throw water on strangers, holidaymakers, foreigners, motorcyclists and buses, anything that moves. Getting soaked under the scorching heat is actually refreshing as it can relieve body heat

 (Source: Songkran) 

Songkran Festival (Source: Wyndham Hollis) 

Along with throwing of water, another thing which people like doing is visiting a 'wat'. Wat is essentially a Buddhist Monastery. Here people pray, give food to monks and give bath to the idols of Buddha.  

songkran (Source: Jimmy theSuperStar) 

Here is a snippet:

Besides the throwing of water, people celebrating Songkran may also go to a wat (Buddhist monastery) to pray and give food to monks. They may also cleanse Buddha images from household shrines as well as Buddha images at monasteries by gently pouring water mixed with a Thai fragrance over them. It is believed that doing this will bring good luck and prosperity for the New Year.

 (Source:  Wikipedia, the free encyclopedia) 

songkran (Source: Jimmy theSuperStar) 

Another way of celebrating this festival is by arranging beauty pageants. Young women show their beauty and talent at these pageants which are judged by the audience. 

Songkran (Source: Jimmy theSuperStar) 

Chiang Mai is the best place in Thailand to participate in Songkran. 

March of the Buddha - Song Kran (Source: Daniel Wabyick)

Here is how people celebrate the three-day festival:

On the first day, firecrackers are let off at dawn and the people of Chiang Mai spring clean their houses. In the afternoon a parade of Buddha images from many Chiang Mai temples goes from the railway station to Wat Phra Singh. People toss lustral water scented with perfume and flowers to bathe the images as they pass along the streets.

On the second day there are no ceremonies but in the afternoon sand is placed in the temple compound as a symbolic return of the sand carried out on the soles of shoes and feet of the people. The sand is made into a small chedi for the next day.

The third day is the start of the New Year and early in the morning is a good time to visit the temples and watch people in traditional costume bringing offerings. Outside people place flags in the sand chedi as well as symbolic sticks of support under the Bo trees to bring good luck. In the afternoon the main government organizations hold a procession from the Yupparat School to honor the Chiang Mai governor at his residence by Nawarat Bridge.

 (Source: Thailand.com)