If your accommodation is around Sukhumvit or Silom, your starting point will be Ratchathewi station (N1), easily reached from the Skytrain Sukhumvit line. Take exit 1 and head a few hundred metres south (after walking down the stairs, turn 180 degrees and walk down the street) and walk over the bridge across the Saen Saeb canal. To your right, you will see the small, rickety pier called Saphan Hua Chang. Board the next canal boat that comes barreling down towards the west (make sure the boat is heading left as you stand on the pier, from under the bridge) — but move quickly, because these boats don't wait for lingerers! Pay your fare (10 baht) and enjoy a view into the backyards of Bangkok. Some points of minor interest that you will pass along the way include:
Jim Thompson House, an attraction in itself and worth a visit if you have some spare energy later;
Bobae Market, one of Bangkok's largest and cheapest clothing markets (not many tourists around here!).
Disembark at the Phan Fa Lilat terminus near the Golden Mount. As in all parts of Bangkok, beware of helpful locals or "tourist police" recommending other sites or advising sites are closed to a certain time! If your accommodation is around Khao San Road, you can just get here on foot. To your left, you'll see a small bridge with oncoming traffic, and a large bridge with multi-lane traffic. Cross the large bridge, pass the white Mahakan Fort and cross the street for Wat Ratchanaddaram, one of Bangkok's unappreciated temples. Entry is free (although a recommended donation of 20 baht is requested), just take your shoes off and climb into the tower for a bird's-eye view of Rattanakosin.
Once finished, head back to the main street and hail a metered taxi (not a tuk-tuk) for a quick trip to Wat Pho; if you insist on the meter, this won't cost you more than about 50 baht. Entry into Wat Pho will set you back 100 baht, with a free bottle of water included. The world's largest Reclining Buddha is in the perennially busy building to your right as you enter from the northern entrance. For good luck, you can do as the Thais do and buy a bag of 25-satang coins to plink, plink, plink into the pots behind the Buddha. Explore the rest of the large complex, and if you feel like taking a break, pop into the famous massage school at the eastern side of the temple grounds.
When done, exit through the west gates (behind the Reclining Buddha) and look for signs pointing towards the pier of Tha Tien, which you will find half-hidden past a market alley full of stalls selling dried squid and such. You will see the white spires of Wat Arun (Temple of Dawn) across the Chao Phraya river. Hop aboard the angular little ferry and pay 3 baht (each way) for the trip. Seeing Wat Arun closer from the gardens around it is free, although quite frankly, it looks better from a distance. You will be charged 50 baht if you want to climb up. Note how the decorations of the spires are in fact all made from millions of bits of smashed white porcelain painted with patterns!
Chao Phraya Express Boat
Head back across the river on the ferry you came on and cross the pier to the Chao Phraya Express Boat dock, clearly identified with blue-and-white signs. Wave down the next southbound (downriver) express boat and ride down all the way to Oriental Pier (N1) for 13 baht. From there, get to the Mandarin Oriental, often judged as the best hotel in the world. Just walk down the alley leading from the dock and turn left. Reward yourself with a drink at the famous Bamboo Bar, although (alas!) live jazz is played only in the evenings. It's expensive though: a beer will cost you a whopping 250 baht. You won't get in if you are wearing sandals either.
From the Oriental, head for the Saphan Taksin BTS station (S6) by either taking the express boat one more stop downriver to Sathorn Pier (Central) or by walking out onto Charoen Krung Road, turning right and walking some 15 minutes south.