Interesting question, though the answer may be found through a number of sources. Let's begin!
Personally, I feel that Taoism and Buddhism are inextricably linked within China. I state this due to the fact that they both rose about the same time, and the fact that they pair well within the Chinese ideology. This is a personal opinion however, and should not be taken as fact. I will write this with this opinion in mind and will detail how both religious belief systems rose and spread throughout China and the world.
Organised Taoism^1 (or Daoism) roughly began within the Han Dynasty (206 BCE - 220 AD) and was formed under the Tianshi Dao^2 sometime around 215 AD. Consequently, Buddhism entered the Han Dynasty through trade with India and other foreign nations around this time. Following the collapse of the Han, China was thrust into an age of disunity (more commonly known as the Three Kingdoms/Northern and Southern Dynasties period; 220-265 AD and 317-589 AD respectively). Keep this in mind as this is crucial in the understanding of why Taoism (and Buddhism) rose after the fall of the Han period!
According to the article The Images of Immortals and Eminent Monks: Religious Mentality in Early Medieval China, the rise of Buddhist and Taoist thought may be attributed to two factors: political unrest and the decline of Confucian thought. Essentially stated, the two religions satisfied the human desire for a means of salvation, and allowed for a more spiritual approach to life's hardships that wasn't present within Confucianism. As previously mentioned, China was at a breaking point prior to the entrance of Taoist and Buddhist thought. What both belief systems did was provide a message of salvation to those suffering following the fall of the Han.
Now, before we continue, one must understand that Chinese dynastic collapse (although similar in some respects) differs greatly from a European collapse. I state this with the following example:
Han Dynasty and the Roman Empire
Similarities
Differences
I bolded this last part as it is CRUCIAL to the spread of both Taoist and Buddhist thought. Simply put, trade allowed both belief systems to spread throughout the dynasties that followed the fall of the Han and the age of disunity. Unfortunately for Taoism however, Buddhism became the dominant belief system of both the Sui and Tang dynasties. All was not lost though, as Taoism remained popular enough (albeit a distinct minority when compared to Buddhism) to spread to Japan within the 7th century (the early Tang Dynasty). I would say this may be attributed to the openness of the Tang Dynasty (still considered the most progressive of the early dynasties) and the fact that the Chinese state had established tributary states within Vietnam, Japan, and Korea. This allowed for Chinese influence to spread to these regions, thereby creating a transfer of ideas between the respective states as emissaries from the tributary states began traveling to China and vice versa.
As for how Taoism spread across the world, the answer is a tad more complex. One may use the above answer of trade to fit nearly every nation state that has seen a rise of Taoist thought, though the rise of western interests within China was surely a factor as well. Along with such, one must understand that the emigration of Chinese nationals who had believed in the Taoist faith would ultimately influence a rise in Taoism within the region in which they settle down. Furthermore, professional Sinologists and Historians with interests in the Far East (specifically China) may have adopted the Taoist belief system as was the case of Henri Maspero. Maspero, a renowned French Sinologist, ultimately wrote nine volumes of Taoist thought, of which would later form the book Taoism and Chinese Religion which seems to have been well received within institutions of higher learning.
There you have it! Keep in mind, I am not an expert Sinologist, nor do I have an extensive background in Far East belief systems. If I am incorrect about any of the above, then I do apologise!
Corrections:
^1 I had previously mentioned that Taoist thought originated in 215 AD. This is incorrect as it dates back to prehistoric China.
^2 I had previously mentioned Lao-tze during 215 AD. This is incorrect.
Sources:
USAF Historian in a rigorous undergraduate program focusing on Far East history and archaeology. Some of this information was taken directly from lecture notes.
The Images of Immortals and Eminent Monks: Religious Mentality in Early Medieval China