Tellurene is a new member of the two-dimensional (2D) materials' family, whose existence has been recently confirmed by first principles calculation and experimental work. Tellurene is also the first 2D mono-elemental material of group-VI predicted by scientists, and investigations of its basic properties are still in their infancy. In this study, we use first principles calculation based on density functional theory to investigate the adsorption of nineteen typical adatoms (Li, Na, K, Ca, Fe, Co, Ni, Cu, Zn, Ag, Au, Pd, Pt, B, N, O, Si, Cl, and Al), and five typical gas molecules (H2, O2, H2O, NO2, and NH3) on α-phase as well as β-phase tellurene sheets. Our calculations shows that most adatoms are chemisorbed on tellurene sheets with large adsorption energies. Moreover, some of the adatoms are observed to give rise to distinct structural deformations and even local reconstructions. We report that a variety of electronic states are induced by the adatoms, which implies that different electronic structures can be engineered by the adsorption of adatoms. In fact, n-type doping, p-type doping, half-metal, and spin-gapless semiconductor features can be acquired by doping adatoms on tellurene sheets. Our calculations also show that the five gas molecules are all physisorbed on tellurene sheets, and no splitting behaviors are observed. Therefore, the adsorption of the five gas molecules has a weak effect on the electronic properties of tellurene. To conclude, our results indicate that adatom engineering may be used to greatly expand the potential applications of 2D tellurene.