The motion of an autophoretic spherical particle in a simple fluid is analyzed. This motion is powered by a chemical species which is absorbed or emitted by the particle and which diffuses and is advected in the surrounding fluid. The transition from the nonmotile to the motile state occurs if the Péclet number Pe (defined as the ratio of the solute emission rate over the solute diffusion rate) is sufficiently large. We first analyze the axisymmetric case (restricting the particle to a unique direction). In this case, we find that the motion of the particle transits from a motionless to a directed motion at a given critical Pe. Increasing Pe, we find a second critical value where the particle becomes stagnant in a symmetric flow. A further increase of Pe leads to a recovery of motile motion. When Pe is increased even further, the particle shows a periodic motion undergoing a subharmonic cascade before entering chaos. In this regime, the mean-square displacement behaves quadratically with time (a ballistic regime). When the axisymmetry constraint is relaxed, allowing the particle to freely move in three-dimensional space, we find that at a small Pe the particle moves in a straight manner. There exists a critical value where the particle exhibits an oscillatory motion with a meandering trajectory. Increasing Pe further leads to chaotic bursts for some time, before entering fully into chaos via the intermittency scenario at a critical Pe number. In this regime, the particle shows run-and-tumble-like dynamics: The trajectory is then characterized by a ballistic swimming nature at a short time and a diffusive nature at a long time.