Miserable in the summer, flooded in the rainy season, and crowded in any season. Bangkok sounds undesirable at the start of any thought, but millions of tourists come anyways. Besides the gritty underbelly, Bangkok has a lot to offer. It is the home to a plethora of regional specialities in Thai food, massages are available at every corner, and thousands of years of history are at your foot.
From the 18th century onwards, Bangkok has been the capital of Siam. The old capital (Ayutthaya) is a day trip away. The old city has limited public transportation, leaving you with TukTuk unless you are already in the old city. On top of the TukTuk, Bangkok traffic is notorious at any time of the day. Maybe it is better to take a ferry or just walk.
The Grand Palace is probably the number one destination for any traveller. Its steep entry will guide you into the massive complex where the monarchy once lived. Numerous ancient and old pantings and gilded Buddhas can be found.
Wat Arun lies across the river, accessible by ferry or TukTuk. It is a great place to see the city as the sun drops or just to get away from the more crowded side of the city. It has a bit of a dreary gray color, but its interesting shape make it more than worth while to visit.
Bangkok has many other temples along the Grand Palace, it is worth just exploring to find out where you will have the best time. One word is to avoid Kaoh San road, which is often picked out as tourist paradise. It is dirty and gritty, plus everyone there is out to cheat unsuspecting tourists. It would not be wise to pay full price for anything there, and the food is mediocre at best.
Silom is far more of a modern district, which also has the Kaoh San drawbacks, but because it is modern, it has larger streets and more comfortable atmosphere. Plus the restaurants offer better food, or at least as good as it gets for a tourist area.