By ship
If you're coming from Rattanakosin, Khao San Road or some areas of Yaowarat and Phahurat, the fastest way to reach Silom is by Chao Phraya Express Boat. A single trip from Chang Pier in Rattanakosin to Sathorn takes about 25 minutes and costs around 18 baht. From here, you can transfer onto the Skytrain's Silom Line and get to many destinations in Silom.
The river boat is the best way to get to and around the Bang Rak area. Si Phraya in the north of Bang Rak and Sathorn in the south are served by fast yellow flag boats. Oriental is the most centrally located, but is only served by orange-flag and no-flag lines. Wat Muang Kae is slightly north of Oriental, but is kind of a sleepy pier only served by no-flag lines.
If you happen to be in Thonburi the area around Khlong San, you can easily take a ferry across the river. There are ferries from Klongsan to Si Phraya, from Wat Suwan to Oriental and from Sathorn across the river to the pier of the same name. Every 15 minutes a ferry leaves and won't cost you more than a mere 3 baht.
By train
Hualamphong Train Station, while technically right over the edge in Yaowarat and Phahurat, is very easy to reach from elsewhere in the district. Just take the metro to Hua Lamphong station and the station will be right in front of you. From there, getting a train is done in no time, and before you know it, you'll be heading for Ayutthaya or Chiang Mai.
By bus
Due to heavy congestion and the complexity of the bus system, getting to Silom by bus can only be advised if you have a lot of time or are on a stringent budget. Being a part of Bangkok's downtown, many lines run through Silom, but finding the right one is a challenge. Ordinary and air-conditioned bus 15 comes from Phra Athit Road and then passes Ratchadamnoen Klang Road for Khao San Road, moves along Lan Luang Road, Chakkaphatdi Phong Road and Bamrung Muang to Siam Square. From there, it continues along Ratchadamri Road and then goes all the way over Silom Road to Charoen Krung Road, the Krung Thep Bridge and The Mall Tha Phra in Thonburi.
From Yaowarat, catch bus 4 from Charoen Krung Road which goes along Rama IV Road, passing Hualamphong Train Station and the Queen Saovabha Institute Snake Farm before reaching the intersection with Silom Road. In the opposite direction, this bus will skip Charoen Krung Road, passing Yaowarat Road instead which is a one-way traffic road.
The first line of the new Bus Rapid Transit BRT system starts at the intersection of Sathorn Road and Naradhiwas Rajanagarindra Road. There is a pedestrian walkway from Chong Nonsi BTS station to Sathorn BRT station, the first stop. From there, the line heads south along Naradhiwas Rajanagarindra Road, then follows Rama III Road along the Chao Phraya River, with Ratchaphruek in Thonburi its final destination. As a commuter line, it is not of much use for foreign visitors, unless you're going to Mambo Cabaret, Tawandang Microbrewery or a distant hotel.