While Internet access using mobile or wireless technologies has become ubiquitous these days, the energy consumption aspect of such connections has not been studied in-depth yet. In this paper, a hybrid wireless network, which consists of a cellular component and a relay-assisted ad hoc component, is studied focusing on energy consumption by mobile stations with respect to the amount of data communicated and achieved battery lifetime. Four alternative paths are considered, including both pure cellular and hybrid ad hoc/cellular links for uplink and downlink traffic. The effects of each alternative connection on energy consumption of the involved mobile stations are analyzed in terms of the amount of data transferred and the operation time of the station's battery. The results from our analysis can also be used for proper relay selection in a hybrid link for achieving optimum data transfer from the Internet while keeping battery energy consumption of the mobile station and/or the relay station at a minimum level.