Magnetic skyrmions are topological quasiparticles with nanoscale size and high mobility, which have potential applications in information storage and spintronic devices. Here, we computationally investigate the dynamics of isolated skyrmions in a ferromagnetic racetrack, where magnetic properties of the edges are enhanced and modified, forming a channel with lower magnetic anisotropy for skyrmion motion. It is found that the rectangular notch at the edge can have a pinning effect on the skyrmion and enrich the dynamics of the skyrmion. Based on the racetrack with modified edges and the notch, we design a racetrack that realizes the skyrmionic logic AND, OR, and NOT gates as well as the diode in the same magnetic racetrack. It is found that the driving current density could be much smaller than those used in previous designs of skyrmion-based logic gates. By slightly altering the shape of the racetrack, we also design the NAND and NOR gates. Finally, we study the feasibility of our design at finite temperatures. Our results may contribute to the design of nonvolatile spintronic devices with integrated multiple functions and ultra-low energy consumption.