We report the discovery of HAT-P-17b,c, a multi-planet system with an inner transiting planet in a short-period, eccentric orbit and an outer planet in a 4.4 yr, nearly circular orbit. The inner planet, HAT-P-17b, transits the bright V = 10.54 early K dwarf star GSC 2717-00417, with an orbital period P = 10.338523 + or - 0.000009 days, orbital eccentricity e = 0.342 + or - 0.006, transit epoch T sub(c) = 2454801.16943 + or - 0.00020 (BJD: barycentric Julian dates throughout the paper are calculated from Coordinated Universal Time (UTC)), and transit duration 0.1690 + or - 0.0009 days. HAT-P-17b has a mass of 0.534 + or - 0.018 M sub(J) and radius of 1.010 + or - 0.029 R sub(J) yielding a mean density of 0.64 + or - 0.05 g cm super(-3). This planet has a relatively low equilibrium temperature in the range 780-927 K, making it an attractive target for follow-up spectroscopic studies. The outer planet, HAT-P-17c, has a significantly longer orbital period P sub(2) = 1610 + or - 20 days and a minimum mass m sub(2) sin i sub(2) = 1.31 super(+0.18) sub(-0.15) M sub(J). The orbital inclination of HAT-P-17c is unknown as transits have not been observed and may not be present. The host star has a mass of 0.86 + or - 0.04 M radius of 0.84 + or - 0.02 R, effective temperature 5246 + or - 80 K, and metallicity Fe/H = 0.00 + or - 0.08. HAT-P-17 is the second multi-planet system detected from ground-based transit surveys.