The fiber geometry, fiber parameters and mode-guiding properties are crucial for realizing high-performance fiber-based sensors. In this work, we propose and demonstrate a few-mode fiber (FMF)-based surface plasmon resonance (SPR) biosensor. The FMF-SPR sensor was fabricated via side-polishing a few-mode fiber and coating a thin layer of gold film, on the basis of the optimization of fiber geometry, thickness of the gold film and mode selection, which were performed with the finite element method. The refractive index (RI) sensing performance of three such sensors with different residual fiber thicknesses were investigated. In the RI range from 1.333 to 1.404, the highest sensitivity up to 4903 nm/RIU and a figure of merit of 46.1 RIU-1 are achieved. For testing the bovine serum albumin (BSA) solution, an averaged BSA RI sensitivity of 6328 nm/RIU and an averaged BSA concentration sensitivity of 1.17 nm/(mg/ml) are realized. Benefiting from only a few modes supported in the FMF, a smaller line-width of the SPR spectrum is obtained, which further results in a higher figure of merit (FOM). Moreover, when combined with the superiority of the mode-multiplexing technology brought by the FMF, the FMF-SPR sensors may find applications in biochemical analysis with high performance and high throughputs.