We present timing and spectral analysis of Swift-XRT (X-ray Telescope) and RXTE-PCA (Rossi X-ray Timing Explorer-proportional Counter Array) observations of the transient Be/X-ray pulsar SWIFT J0513.4-6547 during its outburst in 2009 and its rebrightening in 2014. From 2009 observations, short-term spin-up rate of the source after the peak of the outburst is found to have about half of the value measured at the peak of the outburst by Coe et al. When the source is quiescent between 2009 and 2014, average spin-down rate of the source is measured to be similar to 1.52 x 10(-12) Hz s(-1) indicating a surface dipole magnetic field of similar to 1.5 x 10(13) G assuming a propeller state. From 2014 observations, short-term spin-down rate of the source is measured to be about two orders smaller than this long-term spin-down rate. The orbit of the source is found to be circular which is atypical for transient Be/X-ray binary systems. Hardness ratios of the source correlate with the X-ray luminosity up to 8.4 x 10(36) erg s(-1) in the 3-10 keV band, whereas for higher luminosities hardness ratios remain constant. Pulsed fractions are found to be correlated with the source flux. Overall Swift-XRT and RXTE-PCA energy spectrum of the source fit equally well to a model consisting of blackbody and power law, and a model consisting of a power law with high-energy cut-off. From the pulse phase-resolved spectra and pulse phase-resolved hardness ratios obtained using RXTE-PCA, it is shown that spectrum is softer for the phases between the two peaks of the pulse.