In recent years, more and more scientists have paid more attention to the research on solar activity. There are interannual-century or even millennium scale climate cycles found in the high-resolution paleoclimate proxies data series extracted from the ice cores, stalagmites, tree rings and lake sediments. There are some correlations between the climate changes signals and the solar cycles at the different time scales, however, its physical drive mechanism is still an unsolved mystery. In this study we collected the actual observed and reconstructed data sequences of sunspots data series, and the tree-ring delta~(13)C from the bristlecone pine tree-ring in California (USA) and stalagmite delta~(18)O from the ALH06 stalagmite of Pau d'Alho cave in Brazil and Dongge cave stalagmite in Guizhou of China reflecting monsoon and precipitation intensity,through spectrum analysis,cross spectrum of wavelet transform, coherent spectrum analysis and amplitude modulation. It was found that the periodic signals of climate change recorded in stalagmites and tree rings are significantly correlated with the interannual-millennial scale cycle of solar activity,and that there is a relationship between these periodic signals recorded in the climate proxies and the solar cycles. The lag of 1/4 to 3/4 cycle, according to which it is inferred that the changes in the Earth's surface climate system in interannual-Millennium scale are driven by the solar cycles. In addition, it is found that the period of about 88 a is the amplitude modulation period signal of the sunspot activity cycle of about 11a, while the climate change signals of ca. 500 a, 1000 a and 2000 a are the solar activity cycle ca. 88 a and 210 a amplitude modulation period signals,indicate that the Earth's climate is not only driven by such an eccentricity-modulated precession cycle,but also by the ca. 88 a modulation of the 11a solar activity cycle amplitude, as well as ca. 500 a modulation ca. 88 a period amplitude variation, ca. 1000 a and 2000 a period modulation ca. 210 a period amplitude changes. We found that the Earth is currently in a low amplitude of about 88 a period. The Earth may experience a cooling trend in the next few years, combined with the trend of long eccentricity, which provides an important scientific basis for predicting the short-term and long-term trends for the future global climate change.