Before answering, I should preface with the acknowledgement of the fact that "imperial China" spanned an incredibly long period of time. Due to the constraints with what I feel comfortable with, I'll give a brief note of my understanding of relations in the Tang Dynasty.

Firstly it is worth noting that fundamentally Taoism and Confucianism are far more alike than Buddhism. Confucianism emphasizes family, producing a son, upholding of laws, etc. Buddhists monks (of the era) practice celibacy and considered themselves to be outside of the law. Buddhism started as an outsider religion which proliferated China gradually. It was met quite openly by the populace because of its variability, and Chinese sects developed independently of their counterparts in India, adopting characteristics of Chinese culture (see examples such as the Tiantai School, Huayan School, and Pure Land School). Meanwhile, Confucianism and Taoism both see their origins in early Chinese history. Confucius lived around 500 years B.C.E. (and his teachings were developed by generations of students) and Daoism's principle text, the Daodejing, was written around 300 B.C.E. Thus, they were established ideologies long before Buddhism made its way into Chinese society.

Overall Confucianism is much more concerned with the government, the ruler and his relationship with the people, while Daoism is more preoccupied with the spiritual. Daoism was not so controversial in terms of any conflict with Confucianism, but all three of them were intertwined on way or another. In "China: Prehistory to the Nineteenth Century" J.A.G. Roberts (1) notes that "a scholar might study Confucian texts for his career and read the texts of Daoism for relaxation." and notes that "...many important elements in Daoist teaching were adopted by Confucianism and Buddhism...". This indicates to me that on a day to day level there would not be significant conflicts between followers of each ideology. Indeed, one would expect that many would pick and choose elements and teachings from each which they found relevant. Politically however, there was significant conflict.

In the Tang dynasty, the rise of Buddhist teachings was seen as threatening by early emperors such as Taizong who formerly declared by edict that Daoists monks took "precedence" over Buddhists. Daoists and Buddhist monks would have seen each other as political competitors in the emperor's court, seeking patronage and the construction of monasteries. Political favour wavered between the two sects over time, though Buddhism seemed to be rising in prevalence among the people. This fact and monastic disobedience (both from doctrinal and due to and tax exemption) were what concerned the imperial court. Empress Wu Zetian would swing favour back to the Buddhist clergy, but Emperor Xuanzong would rollback on Buddhism's gains by trying to restrict the monasteries' wealth.

The An Lushan Rebellion made Buddhism even more popular. In part because the imperial court sold ordination licenses to raise funds and ease political pressure. This brings us to an important point. Buddhism was seen by the court (of which Confucian and Daoist scholars were a large part) as an economic problem, though specific emperors had their own personal tastes. Fu Yi (a Daoist priest) is said to have complained that Buddhism was simply a burden to the state. Monasteries paid no taxes, and the state even supported monks and expended significant revenue building monasteries (at least when Buddhism was in the good graces of the current emperor). Some court scholars also disdained Buddhism for its foreign roots. Han Yu (a Confucian scholar) complained that Buddha was a long dead foreigner who understood "neither the duties that bind a sovereign and subject, nor the affections of father and son". In 842 this culminated into the Edict of Suppression of Buddhism (2) by the Wuzong Emperor (who favoured Daoism). He cited reasons such as the foreign nature of Buddhism and accused monks of hoarding wealth from the people. This appears to be a relatively common viewpoint by non-Buddhists of the time. That wasn't the end of the line for Buddhism, as after Wuzong, political favour continued to seesaw, but it should give you some perspective about the political conflict.

You may be wondering about Confucianism at this point. Confucianism does not appear to have been as involved in this political conflict to the same extent as Daoism, though Confucian scholars (such as the aforementioned Han Yu) did oppose Buddhists the same way Daoists did in terms of their arguments. However, Confucianism wasn't vying for patronage so wasn't as direct a competitor. Confucianism was deeply embedded into Chinese culture at this point, and featured in the imperial examination system, which was used to select bureaucrats. Its main function was to help define how governors should act, and many of Confucius's teachings were not so much seen as necessarily "religious" but cultural. Confucianism took a back seat spiritually in the Tang dynasty.

In conclusion, I would say that looking at the Tang dynasty, the relationship between the religions was mostly characterizes by the political conflict between Daoism and Buddhism stemming from competition for patronage. Confucians and Daoists were largely united in seeing Buddhism as an outsider religion that was a drain on national resources (whether or not this was actually the case is less clear).

Hopefully you find this at least somewhat helpful, though this response only seriously covers relations in the Tang dynasty. Perhaps someone else here could help you delve deeper into this topic should you find my answer unsatisfactory.

Research and Sources: (1) "China: Prehistory to the Nineteenth Century" by J.A.G. Roberts. Sutton Publishing (2000) (2) "Edict of the Suppression of Buddhism" http://afe.easia.columbia.edu/ps/cup/emperor_wuzong_suppress_buddhism.pdf