It has been 70 years since the passing of a Thai King, so its impact on modern Thailand is unknown. Most of the information in my answer is based on public announcements made since HM King Bhumipol Adulyadej passed and what occurred after the passing of the Queen Mother several years back.
The official period of mourning will be one year. Government employees will wear black during this time period and the people have also been asked to wear colors of mourning. Colors of mourning here are black or pure white (a pure white shirt not a white guns and roses tshirt). Wearing the colors of the king is also acceptable, but it should be a proper King's shirt, single color with the royal emblem. The King's colors are yellow or pink, the Queen's color is blue. It has not been ordered that people dress accordingly, rather just asked, so you will not get in trouble for wearing green or red, but loud Hawaiian prints and gaudy outfits would be disrespectful. Dress politely and you will be fine.
In addition, a period of 30 days during which flags will be flown and half mast and the people have been asked to "refrain from festivities" will be observed. This may impact tourist's plans, as likely events such as the Full Moon party on Koh Phangan will be canceled and festivals like Loi Krathong and Yee Ping which fall at the end of the 30 day period maybe canceled or subdued and limited.
There are rumors that bars maybe closed, no alcohol, etc, but at this point these are just rumors. There is a strong chance that bars will be asked to shut down music, dancing, etc during the 30 day mourning period. But for now no official bans on alcohol. However I would suggest not walking around with beers in hand, rather enjoy them at your hotel or dining venue. Strolling with a beer in hand would fall into "being festive" and thus disrespectful to the people mourning.
There should be little impact on travel plans and sightseeing activities, though places like the Grand Palace may be restricted to mourners coming to pay respect to the king, likewise at other royal palaces and temples. Bangkok's traffic could worsen over the next couple of weeks, as people from the country side come to pay their respect to their beloved king. But otherwise you should be able to travel freely, sightseeing tours will be running, restaurants, stores, etc will be open as usual.
Bangkok - The roads around the Grand Palace are closed, as is the Grand Palace. Likely nearby temples are over crowded with mourners.
All TV channels, both broadcast and satellite are showing identical programming about the King's life. This is supposed to be in effect for the 3 days according to recent statements by the PM. Internet is uneffected,
Sale of alcohol has been banned for three days, so I assume bars are closed (not in a city at moment to drive around and confirm).
The upcoming Full Moon party has been canceled.
I will edit my answer if any restrictions are announced in the days to come.