This paper studies the distributed formation control problem of second-order multi-agent systems (MASs) with limited communication ranges and collision avoidance constraints. A novel connectivity preservation and collision-free distributed control algorithm is proposed by combining prescribed performance control (PPC) and exponential zeroing control barrier Lyapunov functions (EZCBFs). In particular, we impose the time-varying performance constraints on the relative position and velocity errors between the neighboring agents, and then a PPC-based formation control algorithm is developed such that the connectivity of the communication graph can be preserved at all times, and the prescribed transient and steady performance on the relative position and velocity error can be achieved. Subsequently, by introducing the EZCBFs method, an inequality constraint condition on the control input is derived to guarantee the collision-free formation motion. By regarding the PPC-based formation controller as a nominal input, an actual formation control input is given by solving the quadratic programming (QP) problem such that each agent achieves collision-free formation motion while guaranteeing the connectivity and prescribed performance as much as possible. Finally, numerical simulation is carried out to validate the effectiveness of the proposed algorithm.