Energy harvesting (EH) provides a means of greatly enhancing the lifetime of wireless sensor nodes. However, the randomness inherent in the EH process may cause significant delay for performing sensing operations and transmitting sensed information to the sink. Unlike most existing studies on the delay performance of EH sensor networks, where only the energy consumption of transmission is considered, we consider the energy costs of both sensing and transmission. Specifically, we consider an EH sensor that monitors some status property and adopts a harvest-then-use protocol to perform sensing and transmission. To comprehensively study the delay performance, we consider two complementary metrics and analytically derive their statistics: 1) update age-measuring the time taken from when information is obtained by the sensor to when the sensed information is successfully transmitted to the sink, i.e., how timely the updated information at the sink is, and 2) update cycle-measuring the time duration between two consecutive successful transmissions, i.e., how frequently the information at the sink is updated. Our results show that the consideration of sensing energy cost leads to an important tradeoff between the two metrics: more frequent updates result in less timely information available at the sink.