Ytterbium (Yb) doped optical fibers are widely used in high-power applications and ultrafast lasing since they show adequate power-handling capability and provide desirable beam quality. Yb-doped fibers with large core area can support high power but often act as a multimode fiber and compromise the output beam quality. Hence, it is important to attain a proper balance between the power-handling capability and the beam quality. Yb-doped fibers as a gain medium in pulsed fiber laser systems are prone to nonlinear optical effects due to the presence of high peak power in the ultrashort pulses. Nonlinearity such as self phase modulation (SPM) affects the width and the shape of the pulse, both temporally and spectrally, by inducing chirp during its propagation along the fiber. In this work, finite element method was employed to compute linearly polarized transverse modes and the corresponding modal powers of Yb-fibers with different core areas to optimize the trade-off between the power-handling capability and the beam quality. The optimal fiber was implemented as a gain medium in a passively mode-locked fiber laser system to generate an ultrashort picosecond pulse. The spectral width of the picosecond pulse was studied as a function of pump power to spot the presence of SPM and chirp. To grow a better understanding on the chirped pulse propagation in the fiber, nonlinear Schrödinger equation was numerically simulated in the anomalous dispersion regime under the influence of initial chirp. The simulations reveal a strategy to compress a pulse temporally and utilize it in all-fiber chirped pulse amplification to mitigate nonlinearity in high power applications.