This study presents the synthesis of four series of novel hybrid chalcones (20,21)a–g and (23,24)a–g and six series of 1,3,5-triazine-based pyrimido[4,5-b][1,4]diazepines (28–33)a–g and the evaluation of their anticancer, antibacterial, antifungal, and cytotoxic properties. Chalcones 20b,d, 21a,b,d, 23a,d–g, 24a–g and the pyrimido[4,5-b][1,4]diazepines 29e,g, 30g, 31a,b,e–g, 33a,b,e–g exhibited outstanding anticancer activity against a panel of 60 cancer cell lines with GI50 values between 0.01 and 100 μM and LC50 values in the range of 4.09 μM to >100 μM, several of such derivatives showing higher activity than the standard drug 5-fluorouracil (5-FU). On the other hand, among the synthesized compounds, the best antibacterial properties against N. gonorrhoeae, S. aureus (ATCC 43300), and M. tuberculosis were exhibited by the pyrimido[4,5-b][1,4]diazepines (MICs: 0.25–62.5 µg/mL). The antifungal activity studies showed that triazinylamino-chalcone 29e and triazinyloxy-chalcone 31g were the most active compounds against T. rubrum and T. mentagrophytes and A. fumigatus, respectively (MICs = 62.5 μg/mL). Hemolytic activity studies and in silico toxicity analysis demonstrated that most of the compounds are safe.