This is a huge question and involves enormous amounts of records. But I'll quickly summarize an article on the nationalist army's retreat to Taiwan in 1949, published by the National Archives Administration, National Development Council of Taiwan.

Source (Mandarin) with images of official records of the retreat: http://www.archives.gov.tw/Publish.aspx?cnid=948&p=1247

Since 1949, with the ultimate goal of winning the civil war, the nationalist government decided to temporarily retreat from the mainland, setting up shop in Taiwan to give the battered army a break and rebuild. But due to changes in leadership, and a split on opinion regarding whether to keep on fighting or negotiate peace, there were no grand plan for retreat. Differnet branches of the armed forces had their own plans, making it a very disjointed and chaotic process.

There's not much to say about the air force, they flew over.

The navy started planning their retreat in 1938 after the Huaihai Campaign in which the communist army obliterated 56 Divisions of nationalist forces. Over the next two years, the navy assisted the army in transporting the land troops from the eastern coast of China to Taiwan.

The army was much more complicated, since they had the communist army on their tail. Many units were simply caught and eliminated by the communist army, many surrendered, with relatively few successfully arriving in Taiwan. Estimates of the number of actual military personnel that made it to Taiwan usually fall at between 500,000-600,000 men.

There were also units that coulnd't get to the coast but refused to surrender, ending up retreating south-westerly and hid in the jungles by the border of Thailand and Myanmar. The earliest group of these consisted of 1,500 men who planned to reach Taiwan via Thailand, but they had to ditch the plan for being too difficult. Within 2 months, their numbers grew to 3,000 men and dug in. After refusing Myanmar's demands for them to leave, they defeated the assaulting Myanmar troops, gaining international publicity. In the end, under international pressure, KMT agreed to withdraw those troops. However, part of the troops refused to leave, instead settling down in northern Thailand. They later joined the Thai army and were sent to eliminate the remnants of Thai communist elements with heavy losses. For their efforts, the King of Thailand granted audience to their commander and gave all of them the rights to reside in Thailand. Now they still have close connections with Taiwan and their children often come to Taiwan to study and work.

Back to the bigger picture. In 1949, the nationalist government had already ordered various branches of the armed forces to set up a Taiwan defense plan.

In the latter half of 1949 when the communists had held over half of the mainland, the nationalist government started transferring all the administrative branches (and their personnel) to Taiwan. By the end of the year, they announced to move the capital to Taipei.

So while it was a hurried run, it wasn't just something spontaneous like how people would run from a sudden earth quake. The retreat spanned a year or more, with the navy and air force providing transport for army personnel and civilians. All the while the army was still trying to hold off the communist troops with little effect.

The famous National Palace Museum, for example, already started selecting and packing its collections in 1948, with the first batch arriving in Taiwan by December.

Lastly, while politicians (for example the representative in the National Assembly) generally had connections and got out on military transports along with government employees, regular citizens were not that lucky. They had to pay an arm and a leg for tickets to the many freighters that traveled between Taiwan and China. A film based on one of the steamer that sunk is The Crossing.