Getting into Rattanakosin is quite a hassle, as the Skytrain and metro systems do not cover the area not yet, at least. For time being, your options are to either take a taxi, or use the boats:
By bus
While not as chaotic as downtown, bus travel here is still not a good idea. However, as other good public transport options are limited, bus travel can be the fastest way of travel from downtown though this is heavily depending on traffic. The road around Sanam Luang is one of the city's bus hubs, but as it is a circular road, it is quite hard to find out where the bus is going. It is a better idea to get on the bus in front of Tha Chang Pier near the Grand Palace.
Ordinary bus 47 starts at Tha Tien Pier near Wat Pho and then passes Tha Chang Pier, Ratchadamnoen Klang Road for Khao San Road, Lan Luang Road, Chakkaphatdi Phong Road, Bamrung Muang Road and finally reaches MBK Center for Siam Square. It then continues its way along Phaya Thai Road and Rama IV Road before reaching the intersection with Silom Road.
Another option is ordinary and air-conditioned bus 25, which starts at Tha Chang and then continues to Charoen Krung Road. In opposite direction it will skip Charoen Krung Road and take Yaowarat Road instead. Then it continues its way along Rama IV Road passing Hualamphong Train Station, Queen Saovabha Institute Snake Farm and the intersection with Silom Road. Then it heads north through Ratchadamri Road to Ratchaprasong intersection for Siam Square, from which it heads all the way southeast along Phloen Chit Road and Sukhumvit Road.
By ship
The best way to get in Rattanakosin is by Chao Phraya Express Boat. Besides a relatively quick and convenient way of travel, it is also a much more enjoyable than looking at cars stuck in traffic while inhaling exhaust gases. The wind from the river is a relaxed way to cool down from tropical temperatures, too.
If you're coming from downtown, you can take the Skytrain to Saphan Taksin station and transfer onto the express boat. A single trip from Saphan Taksin to Tha Chang near the Grand Palace takes about 30 minutes and costs 15 baht. Other notable stops include Tha Tien for Wat Pho, Phra Arthit for Khao San Road and Thewet for Dusit. It is best to take the orange flag boats, as yellow flag boats miss out on many important stops.
From Tha Tien Wat Pho, you can take a ferry shuttle to Wat Arun, across the river at the Thonburi side. Ferries leave every 10 minutes for just 3 baht. As many students from the universities live across the river in Thonburi, there are plenty of other commuter ferries that cross the river. Tha Chang has ferries to Wat Rakung and Wang Lang Siriraj, Tha Phra Chan has ferries to Wang Lang Siriraj, Thonburi Railway and Phra Pin Klao Bridge, and Thewet has a ferry to Wat Kharubodj.
Eastern Rattanakosin roughly the area around the Golden Mount is served by the Golden Mount Line of the Saen Saep Express Boat service. If you happen to be in that area, using the canal boat is the quickest way to get to Siam Square, Sukhumvit and Ramkhamhaeng. The stop closest to the Golden Mount is Panfa Leelard, from which there are direct services to Pratunam. There you could switch onto the NIDA Line, that runs from Pratunam all the way northeast to Wat Sriboonreung in Ramkhamhaeng. A single trip from Panfa Leelard to Pratunam takes about 20 minutes and costs 12 baht. On the trip, you will pass the stops Talad Bobae for the garment market of the same name, Sapan Charoenpol, Baan Krua Nua for Jim Thompson's House, Sapan Hua Chang for Siam Square and Pratunam for Pratunam and Ratchaprasong.