A quick glance at a map of Sevilla will show you a city divided practically right down the middle by the Guadalquivir River. You'll find the majority of the action on the eastern bank in the barrios (neighborhoods) of La Macarena, El Arenal, the centro, Santa Cruz, and María Luisa. However, don't miss the western bank barrios, especially Triana, as they offer up a less-touristy and certainly unique atmosphere of their own.
The most enjoyable and picturesque way to keep in shape without emptying your wallet is a nice, healthy jog in the great outdoors! The flat streets of Sevilla provide for great jogging options, but there are plenty of places ideal for a nice jog away from the noise and the slow-moving crowds. Sevilla's parks are so beautiful that you won't even pay attention to that nagging cramp in your side, while the breeze coming off of the Guadalquivir will keep you refreshed as you trod along its shore-hugging path. 
