With the passing of Thai King Bhumibol, are there any customs/etiquette as a traveler I should be aware of?

An official statenent from Tourism Authority of Thailand is as follow

Thailand is now in an official period of mourning following the passing of His Majesty King Bhumibol Adulyadej.

In this regard, the Tourism Authority of Thailand (TAT) would like all visitors to Thailand to continue with their travel plans as normal. Furthermore, kindly be advised of the following:

• Many Thai people will be wearing black or white clothing as a sign of mourning. This is not required of visitors but if possible, they should wear sombre and respectful clothing when in public.

• Visitors should refrain from conducting any inappropriate or disrespectful behaviour.

• Tourist attractions will be open as usual with the exception of Wat Phra Kaeo (Temple of the Emerald Buddha) and the Grand Palace, as they will be the venue of the Royal Funeral Rites.

• The Government has asked for the cooperation from the entertainment venues; such as, bars and nightclubs to consider the opening of their business operations during this time. The decision will be made by the individual owners.

• Most of the traditional and cultural events will be taking place as usual, although the celebrations may be changed for appropriateness as a mark of respect, or the events may be dedicated to the memory of His Majesty King Bhumibol Adulyadej.

• All transport, banks, hospitals and other public services will be operating as usual.

• The related authorities have stepped up safety and security measures for all Thais and visitors to facilitate their travelling around the country.
• For any enquiries, please contact our TAT domestic offices or Tel.: 1672.

TAT would like to thank all visitors for their understanding and supporting Thailand during this period of time.

Nightclubs and bars are allowed to open as usual but all entertainment must perform in indoor place (no full moon party on the beach etc). Places that is strictly closed for all visitors is Royal Palace and Temple of Emerald Bhuddha. Other than that plan your trip as normal, more time for heavier traffic jam and bring lots of dark-color clothes for occasion.

You no need to worry about travelling to Thailand this time.
Thai people always welcome all tourists even a hardly time at this moment.
All tourist places open as normal, except Royal Grand Palace will close till 20 Oct 2016.
Only just wear non brightly colored clothes to pay respect for all Thais and His Majesty of King Bhumibol Adulyadej.

“Should I be aware of any customs/etiquette as a traveler when visiting Thailand?”
I think, being in a good manner will work for any country, any occasion.

“the entire country will be in mourning for a year”
from what I read and ask thai friends, it is for government officers and public servants. Private business was asked for corporation for 30 days.

Back when they have funeral for the King’s mom and later his aunt, they also open for public to attend the funeral. I am sure, they will do the same for the King funeral. You should attend the funeral with the locals if you can. It will be one in a lifetime experience.

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.

Updates

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.

One must understand that Thai citizens are very sensitive about the loss of the King. Unlike political parties, where controversy is normal, HM King Bhumibol was sincerely beloved by everyone in the country.

As a foreigner, your goal can be stated in one phrase: don’t make it worse. The reason is simple: many your actions that could be tolerated in a normal situation, may cause you trouble these days.

• Etiquette. You may expect many locations (more than usual) decorated with Thai flags and pictures of the King. Watch your gestures, don’t point at these as someone may see this as disrespect (even though you may be talking about something else at this time)
  Doing something incorrectly is worse than doing nothing. If you want to wai to a portrait of the King, make sure you know how to do it.
• Dress. I haven’t seen any official recommendations (yet), but jpatokal’s answer seems to be quite good.
• Political. I would recommend refraining from any kind of political discussions in Thailand. Besides the Lèse-majesté law that prohibits offending the dignity of a reigning sovereign, don’t even discuss the political parties of Thailand.
• Entertainment. There is no detailed information yet from Thai officials, but one may expect the entertainment zones to be shut down for a considerable period. If your travel is for entertainment purposes, please reconsider your visit.
• Temples. If the purpose of your travel is visiting temples, expect large crowds, as many people will go to temples to pray for the King, especially in densely-populated areas.
• Police. Some areas can have restricted access. If you see armed police or security blocking some area/building, make sure you are still allowed to enter even if there’s no physical barrier. In case of any doubt, politely ask a policeman.

It’s my personal opinion that if your planned visit to Thailand can be rescheduled you should do so.

Not in particular. However, in the immediate mourning period, expect to see a lot of nightlife (clubs, go-go bars etc) closed, many celebratory events (parties, concerts etc) cancelled and possibly some restrictions on alcohol sales.

That said, the monarchy has always been a touchy topic in Thailand, but it’s become particularly touchy under the current junta and will be even more so for a while. It’s best to avoid the topic entirely and restrict yourself to vague condolences if it does come up.

Finally, some colour codes to be aware of: the King’s colours are yellow and pink. Yellow in particular is associated with the royalist faction in politics (currently in power, although the original “yellow shirts” have mostly been discredited; it’s complicated), while pink and the light blue associated with Queen Sirikit are more neutral. Expect to see a lot of people wearing these for a while. Red, on the other hand, is associated with deposed Prime Minister Thaksin’s party and the opposition. Foreigners aren’t expected to know these nuances, so you’re not going to get beaten up for wearing a Manchester United shirt, but there’s no harm in sticking to neutral colours.