The Banglamphu district is located to the east of Ko Rattanakosin, and there are quite a few decent sights round here. It's not well defined where Banglamphu actually starts and finishes, and some sights included here such as Wat Saket and Wat Suthat are sometimes considered to be outside Banglamphu.
About 10 minutes walk north of Sanam Luang (in Ko Rattanakosin), is Khao San road. This is home to a very lively travelers scene, and a mass of budget accommodation is here. Some love it, some hate it but it's worth walking down even if you don't plan to stay here. A extensive market with a huge range of goods, from cheap clothes to Thai handicrafts, many inexpensive restaurants and all the different types of people that come here can make it a interesting experience. Somewhat surprisingly, the nearby backstreets retain a genuinely Thai feeling, much more so than Bangkok's other main tourist accommodation areas, Sukhumvit and Silom. The start of the film The Beach was also set in Khao San road, though filmed in Manila as the real Khao San was judged to be not 'realistic enough'. The guesthouse rooms shown in the film are accurate enough, but if you come expecting to be challenged to drink snake's blood as you walk along the street, you'll end up disappointed.