Prosody contains rich information beyond the literal meaning of words, which is crucial for the intelligibility of speech. Current models still fall short in phrasing and intonation; they not only miss or misplace breaks when synthesizing long sentences with complex structures but also produce unnatural intonation. We propose ProsodyFM, a prosody-aware text-to-speech synthesis (TTS) model with a flow-matching (FM) backbone that aims to enhance the phrasing and intonation aspects of prosody. ProsodyFM introduces two key components: a Phrase Break Encoder to capture initial phrase break locations, followed by a Duration Predictor for the flexible adjustment of break durations; and a Terminal Intonation Encoder which learns a bank of intonation shape tokens combined with a novel Pitch Processor for more robust modeling of human-perceived intonation change. ProsodyFM is trained with no explicit prosodic labels and yet can uncover a broad spectrum of break durations and intonation patterns. Experimental results demonstrate that ProsodyFM can effectively improve the phrasing and intonation aspects of prosody, thereby enhancing the overall intelligibility compared to four state-of-the-art (SOTA) models. Out-of-distribution experiments show that this prosody improvement can further bring ProsodyFM superior generalizability for unseen complex sentences and speakers. Our case study intuitively illustrates the powerful and fine-grained controllability of ProsodyFM over phrasing and intonation.