The Grand Palace is a huge royal ground, and a strict dress code applies.
The first attraction is immediately the most important one, the gigantic Grand Palace. If your accommodation is in Sukhumvit or Silom, take the BTS Skytrain to Saphan Taksin. Then take the Chao Phraya Express Boat north up to Chang Pier, which will take around 30 minutes.
When disembarking walk left to Santichaiprakarn Park. If you live around Khao San Road, you can just walk through Soi Rambuttri, then take a right through Soi Chana Songkram. From there, nearby is a little alley to Phra Athit Pier (N13). Take the Chao Phraya Express Boat to Chang Pier, which will take about 10 minutes. If you have a lot of energy, you could even walk the whole way to the Grand Palace, which will take around 35 minutes.
From Chang Pier, walk straight onto Na Phra Lan Road, and you will see the white Grand Palace fortress walls. It's not easy to find your way in, but just keep on walking till you see the gigantic Ministery of Defense. Check out the City Pillar Shrine (Lak Muang) first and then head your way to the entrance. The most important part is Wat Phra Kaeo (Temple of the Emerald Buddha). You will be amazed by all the golden shiny buildings. Admission is B500.
When done, it's probably time for lunch. There are barely any classy restaurants here, so it's best to eat the way the locals do. A mid-range restaurant on a short walk from the Grand Palace is Elle Tha Phra Chan, opposite from Silpakorn University at Maharat Road. Order some of Thailand's most typical dishes, like Tom Yam Goong, Tom Kha Kai or Pad Thai, for about B70 each. Budget travelers can eat plenty of cheap rice and meat dishes for B40 at the Tha Chang Market. It's a nice food market with plenty of fruit to choose from. It's also a nice place if you're just looking for a quick snack, like meat on a stick or Takoyaki.
Head up north onto Maharat Road, which is an interesting Amulet Market. If you want to get lucky, buy an amulet for as cheap as 5 baht. Besides the amulets, some remarkable products are for sale here, such as ready-to-use dentures! Walk onto Phra Chan Market, which is mostly a clothes market. Most popular here are the t-shirts in pop-art style.
If you're up for the National Museum, walk into Phra Chan Road and take a right at the massive Na Phra That Road. The museum is a good introduction into Thai history, culture and religion. Admission is 40 baht, expect to be here at least two hours. You can also skip it, and go to the National Gallery later.
When done, walk back to the Phra Chan Market, and take a right into the Thammasat University complex. Keep on walking till you leave the complex and get onto 16th August Road. You might want to sip a cup of coffee at Ta Chang Cafe, or just walk on to the Bangkok Information Office under the Phra Pinklao Bridge. As it's probably your first day in town, get some leaflets and detailed maps of the area. If you want to visit the National Gallery, walk into Chao Fa Road and it will be on the left in about 5 minutes. If not, walk past the Old City Moat and onto the boardwalk. It's a nice and breezy walk along the Chao Phraya river.
After about 10 minutes, the boardwalk will lead you to Santichaiprakarn Park. As the sun is blisteringly hot, just take some time off here in the shadow with a cool beverage. Check out the Phra Sumen Fortress and enjoy the jugglers and other performers in the park. If you want to take the Chao Phraya Express Boat back to your accommodation, you will have to do it before 5.30PM. If not, your evening will be centered on Khao San Road.
Khao San Road is the backpackers mecca of the city. Tourists are everywhere, but also locals go here for a good night out. To get there from the park, walk onto Phra Athit and take a left onto Soi Chana Songkhram, the little alley next to 7 Eleven. You can get a Thai massage around here, or try to buy some clothes or books you have been looking for. When you walked through Soi Rammbutri, take a right and make a short walk on the big Chakra Bongse Road. On the right side, you will see Wat Chana Songhkram. Be careful when crossing the street to Khao San Road.
Khao San Road is a great place for eating cheap dinner. One of the best medium priced restaurants in the area is Tom Yum Kung Restaurant. It might look sleazy when judging from the big neon sign outside, but it's square is actually very classy in traditional Thai style. An obvious treat here is Tom Yum Goong, but try some of their delicious seafood. If you're on a budget, just go to one of the carts selling Pad Thai for B50.
Plenty of locals go to Khao San Road just for it's nightlife. There are plenty of sports bars, Irish pubs and clubs here, so there must be something of your liking. A popular place is Brick Bar, in the alley behind McDonald's (where you can also find the toilets). Be sure to bring your passport with you, as identification is compulsory. Inside, you'll find an old fashioned Western style bar with performing bands.
Families could sit outside in one of the numerous sidewalk cafes. Recently opened is Mulligans Irish Pub, where you can chill outside and drink a beer for B80. It's a good place for conversations, as they do not have any loud music outside. Yet, the bar at the other side does have loud music, so the atmosphere is still fine. Khao San Road is a great place for just watching people go by.
Getting back to Sukhumvit or Silom at night is a bit of a pickle. Your best bet is to take a metered taxi, which will cost you around B150.