The Internet is increasingly used by young children for all kinds of purposes. Nonetheless, there are not many resources especially designed for children on the Internet and most of the content online is designed for grown-up users. This situation is problematic if we consider the large differences between young users and adults since their topic interests, computer skills, and language capabilities evolve rapidly during childhood. There is little research aimed at exploring and measuring the difficulties that children encounter on the Internet when searching for information and browsing for content. In the first part of this work, we employed query logs from a commercial search engine to quantify the difficulties children of different ages encounter on the Internet and to characterize the topics that they search for. We employed query metrics (e.g., the fraction of queries posed in natural language), session metrics (e.g., the fraction of abandoned sessions), and click activity (e.g., the fraction of ad clicks). The search logs were also used to retrace stages of child development. Concretely, we looked for changes in interests (e.g., the distribution of topics searched) and language development (e.g., the readability of the content accessed and the vocabulary size). In the second part of this work, we employed toolbar logs from a commercial search engine to characterize the browsing behavior of young users, particularly to understand the activities on the Internet that trigger search. We quantified the proportion of browsing and search activity in the toolbar sessions and we estimated the likelihood of a user to carry out search on the Web vertical and multimedia verticals (i.e., videos and images) given that the previous event is another search event or a browsing event. We observed that these metrics clearly demonstrate an increased level of confusion and unsuccessful search sessions among children. We also found a clear relation between the reading level of the clicked pages and characteristics of the users such as age and educational attainment. In terms of browsing behavior, children were found to start their activities on the Internet with a search engine (instead of directly browsing content) more often than adults. We also observed a significantly larger amount of browsing activity for the case of teenager users. Interestingly we also found that if children visit knowledge-related Web sites (i.e., information-dense pages such as Wikipedia articles), they subsequently do more Web searches than adults. Additionally, children and especially teenagers were found to have a greater tendency to engage in multimedia search, which calls to improve the aggregation of multimedia results into the current search result pages.