The temporal relationship between coronal mass ejections (CMEs) and associated solar flares is of great importance to understanding the origin of CMEs, but it has been difficult to study owing to the nature of CME detection. In this paper, we investigate this issue using the Large Angle and Spectrometric Coronagraph and the EUV Imaging Telescope observations combined with GOES soft X-ray observations. We present four well-observed events whose source regions are close to the limb such that we are able to directly measure the CMEs' initial evolution in the low corona (~ 1-3 R) without any extrapolation; this height range was not available in previous space-based coronagraph observations. The velocity-time profiles show that kinematic evolution of three of the four CMEs can be described in a three-phase scenario: the initiation phase, impulsive acceleration phase, and propagation phase. The initiation phase is characterized by a slow ascension with a speed less than 80 km s^-1 for a period of tens of minutes. The initiation phase always occurs before the onset of the associated flare. Following the initiation phase, the CMEs display an impulsive acceleration phase that coincides very well with the flares' rise phase lasting for a few to tens of minutes. The acceleration of CMEs ceases near the peak time of the soft X-ray flares. The CMEs then undergo a propagation phase, which is characterized by a constant speed or slowly decreasing in speed. The acceleration rates in the impulsive acceleration phase are in the range of 100-500 m s^-2. One CME (on 1997 November 6, associated with an X9.4 flare) does not show an initiation phase. It has an extremely large acceleration rate of 7300 m s^-2. The possible causes of CME initiation and acceleration in connection with flares are explored.