Many societies are organized in networks that are formed by people who meet and interact over time. In this paper, we present a first model to capture the micro-foundations of social networks evolution, where boundedly rational agents of different types join the network; meet other agents stochastically over time; and consequently decide to form social ties. A basic premise of our model is that in real-world networks, agents form links by reasoning about the benefits that agents they meet over time can bestow. We study the evolution of the emerging networks in terms of friendship and popularity acquisition given the following exogenous parameters: structural opportunism, type distribution, homophily, and social gregariousness. We show that the time needed for an agent to find "friends" is influenced by the exogenous parameters: agents who are more gregarious, more homophilic, less opportunistic, or belong to a type "minority" spend a longer time on average searching for friendships. Moreover, we show that preferential attachment is a consequence of an emerging doubly preferential meeting process: a process that guides agents of a certain type to meet more popular similar-type agents with a higher probability, thereby creating asymmetries in the popularity evolution of different types of agents.