Influenza surveillance has typically been carried out using influenza-like illness (ILI) rates and proportions of laboratory tests positive for influenza as metrics to monitor, with sample sizes for the number of tests to carry out based on the precision of the resulting estimate of proportions positive. The transition out of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) pandemic period has encouraged the establishment of integrated surveillance of respiratory pathogens, in the context of multiple surveillance objectives, as set out by WHO in its revised integrated surveillance guidance and Mosaic Respiratory Surveillance Framework. These objectives include outbreak detection, situational awareness and intensity evaluation, among others. We illustrate how to design respiratory surveillance in primary care, by considering multiple surveillance objectives for different metrics of different types of respiratory pathogen circulation seasons in England, the USA and Hong Kong. We focus on a proxy of influenza activity as a metric to compare between these countries/regions. Taking advantage of England's integrated sentinel primary care surveillance system, we propose further metrics to monitor: a proxy of respiratory activity, novelly defined as the product of an acute respiratory infection (ARI) consultation rate and the proportion of tests positive for emph{at least one pathogen}; pathogen-specific ARI-based activity proxies for more detailed monitoring of influenza and SARS-CoV-2; and integrated monitoring of proportions positive for all pathogens tested. We use a simulation approach to determine sample sizes by optimising either the probability of, or time to, detection of different events in monitored metrics, according to the different surveillance objectives. We find that sample sizes to maximise detection probabilities or minimise detection times vary by metric, objective, event and country/region. At a national level, the current sample sizes used are sufficient to detect most events in most weeks for both the USA and Hong Kong, but for England the numbers of swabs taken for ILI consultations may not be sufficient in all weeks, particularly at the start of the season when outbreak detection is important. However, broadening the criteria for swabbing to acute respiratory symptoms does allow for sufficient sample sizes.