Autonomous fabric manipulation is a challenging task due to complex dynamics and self-occlusion during fabric handling. An intuitive method of fabric folding manipulation involves obtaining a smooth and unfolded fabric configuration before the folding process begins. To achieve this goal, we propose an efficient dual-arm manipulation system that combines quasi-static (including pick & place and pick & drag) and dynamic fling actions to flexibly manipulate fabrics into unfolded and smooth structures. Once this is done, keypoints of the fabric are detected, enabling autonomous folding. We evaluate the effectiveness of our proposed system in real-world settings, where it consistently and reliably unfolds and folds various types of fabrics, including challenging situations such as long-sleeved T-shirts with most parts of sleeves tucked inside the garment. Our method exhibits high efficiency in fabric manipulation, even for complex fabric configurations. Our method achieves a coverage rate of 0.822 and a success rate of 0.88 for long-sleeved T-shirts folding.