Bowing
One of the most obvious social conventions is the bow. Everyone bows when they say hello, goodbye, thank you or sorry. Bowing is a term of respect, remorse, gratitude and greeting.
If you meet someone in Japan you may wish to give them a little bow, but you do not necessarily need to bow to everyone who bows to you. Entering a shop or restaurant for example, you will be greeted by shouts of irrashaimase (welcome) and a bow from the staff as a sign of respect to you as the customer.
As the customer, you will not be expected to bow back as you could be facing a long bow-off as the staff will feel it necessary to bow back to you. You may prefer to adopt the casual head-nod version of the bow as a sign of acknowledgement when thanked for your purchase at the end of your shopping experience. Many Japanese people use the head-nod in more casual everyday situations.
There are several forms of bowing, such as the 45-degree saikeirei bow used for moments for sincere apology or to show the highest of respect, or the 30-degree keirei bow, which is also used to show respect to superiors. As a visitor to Japan you will probably have no use for either of these.
The eshaku 15-degree bow is semi-formal and used for greetings when meeting people for the first time. You may have more use for this bow during your time in Japan, but you will not be expected to use it and Japanese these days are more than familiar with shaking hands.