Chinatown is a great part of town for photos. The main street, Yaowarat, can be rather crowded, but the smaller streets are more quiet. The best ways to get there is by boat from Saphan Taksin (sky-train station with the same name) to Rachawong or take the MRT (underground) to Hua Lampong (the main railway station) and walk from there.
Another interesting area is Pratu Nam, particularly the clothing market. Much of the market has been modernised though, and the dark, old, dusty places can be a bit hard to find.
From Pratu Nam, you can take a canal boat to the Golden Mount (Wat Saket) which was mentioned by Jeff above. The Golden Mount is at the last station, so you don't need to worry about where to get off. If it's not too hot, Grand Palace and Wat Prakaew is within 30 minutes walking distance from the Golden Mount along Rachadamnoen, the parade street of Bangkok. Just be sure not to go down Rachadamnoen Nok which starts at the same location but goes in more or less the opposite direction.
Another temple near Sanam Lueang (the location of Grand Palace etc.) is Wat Mahathat. A very beautiful temple and much more tranquil than the crowded Wat Prakaew. North of that temple there's a street called Phra Chan. If you walk that street down to the river, there's a ferry pier, but through a small passage to the left, there's a narrow "footpath" along the river with small, local restaurants. It can be a bit flooded now during the rainy season and it's very crowded at lunch time (12.00-13.00), but it's a nice place for a mid-day break.
If you take the small ferry across the river to the Sirirat Hospital, there are also riverside restaurants and a nice walk down towards Wat Arun. The best sunset view to Wat Arun is from the other (East) side of the river, a restaurant/bar called Deck by the River at 36-38 Soi Pratoo Nok Yoong, Maharat Road. The outdoor bar on the top floor has a perfect view to the temple, but is obviously also very popular.