In order to investigate the different aspects of the transport of heat in the absence of gravity, we performed a thermal cycle close to and around the critical point of CO2 at critical density. Reduced gravity was provided during a 6-min flight of a sounding rocket. A cell has been designed which allows surface and bulk phenomena to be distinguished. Special optical devices are used to observe the sample and measure locally temperature and density. We also present experiments under Earth's gravity at a few mK from the critical temperature (T(c)). Convection patterns are observed which correspond to Grashof numbers as large as a few 10(4). The thermal cycle of the experiment in weightlessness starts at T(c) +2.5 mK, where we study the relaxation of the perturbations caused by liftoff: fluid flows, density and temperature gradients. We then investigate the effect of a quench from T(c) + 2.3 to T(c) + 1.3 mK, and the expected mechanism of heat transport by the ''Piston effect'' (PE) is experimentally evidenced. In particular, we observe homogeneous thermalization, with a time of thermalization less than a few seconds. We compare this time with a number of theoretical estimations. During thermalization, density and temperature gradients are not affected by the PE and are seen to relax diffusively. We perform then a quench at T(c) - 0.8 mK. We observe the onset of phase separation, making a clear visualization of the PE. Thereafter, spinodal decomposition occurs, followed by the growth of a bicontinuous pattern of gas and liquid domains. After a last quench back to T(c) + 2.0 mK, we still observe the PE in spite of the presence of density gradients. These gradients are seen to relax by diffusion.