We present the results of our analysis of the photospheric and chromospheric activity of the star DS Tuc (DS Tuc A, HD 222259A), which is a member of the Tucana-Horologium (Tuc-Hor) group with an age of 45 Myr. The star DS Tuc may be considered as the prototype of a young Sun, while its planetary system may be considered as the prototype of a young Solar system. Using data from the archive of TESS observations, we have analyzed the activity of this star. According to our estimate, the stellar rotation period is $P=2.85± 0.18$ days, which coincides with its known estimates. The light curve has been analyzed by the method of solving the inverse problem of reconstructing the maps of surface temperature inhomogeneities. We show that there are concentrations of spots at two longitudes on these maps. The spot positions undergo changes. The fraction $S$ of the spotted stellar surface is about 3.3 $%$. We have studied the object's position on the $S$-age, $S$-rotation period, and $S$-Rossby number diagrams and concluded that it corresponds to the general pattern of the dependences established previously for young stars with planetary systems and solar-type stars. Based on All Sky Automated Survey observations, we have estimated for the first time the possible activity cycle of DS Tuc to be 1610 days (4.4 years).