SRGA J144459.2$-$604207 is a newly confirmed accreting millisecond X-ray pulsar and type I X-ray burster. We present the broadband X-ray timing and spectral behaviors of SRGA J144459.2$-$604207 during its 2024 outburst. The data were collected from NICER, Einstein Probe, IXPE, Insight-HXMT, NuSTAR and INTEGRAL observations. X-ray pulsations have been detected for the 1.5--90 keV energy range throughout the `ON' phase of the outburst from MJD $\sim 60355-60385$. We refined the orbital and spin ephemerides assuming a circular orbit, and found that the pulsar was in a spin-up state during MJD $\sim$ 60361--60377 showing a significant spin-up rate $\dot{\nu}$ of $(3.15\pm 0.36)\times10^{-13}~{\rm Hz~s^{-1}}$. Around MJD $\sim 60377$ a swing was detected in the spin evolution accompanied by significantly enhanced pulsed emission. We studied the pulse profile morphology during the X-ray bursts as observed by Insight-HXMT, IXPE and NuSTAR. During the bursts, pulsations were detected across the 2--60 keV with shapes broadly consistent with those observed for the persistent emission. We found, however, that the `burst' pulse profiles exhibit significant phase offsets relative to the pre- and post-burst profiles. These offsets systematically decrease with increasing energy, $\Delta \phi\approx0.15$, 0.11 and 0.02 for IXPE, Insight-HXMT ME and HE in 2--8, 5--30 and 20--60 keV, respectively, and $\Delta \phi \approx 0.21$, 0.10 and 0.07 for NuSTAR in 3--10, 20--35 and 35--60 keV, respectively, compared to the pre- and post-burst profiles. We performed a joint spectral analysis of quasi-simultaneous NICER, NuSTAR, and Insight-HXMT data for two epochs. The resulting spectra from both observations were consistent and well-described by an absorbed thermal Comptonization model, nthcomp, plus relativistic reflection, relxillCp.