Most people are familiar with Asakusa Kannon Temple (Sensoji) and it's famous Kaminarimon Gate and Nakamise Dori Street. Inside the grounds of the temple is a 5 story pagoda, Hozomon Gate, the Asakusa Shrine, Nitemmon Gate,... Standards for all visitors to Asakusa.
What is possible to miss is what surrounds the temple. Hours and hours can be spent wandering the many narrow alleys and shopping arcades. This is the Asakusa that I recommend most.
Pick up a map at the information center across the street from the Kaminarimon Gate. Concentrate on the section that is located to the left of the temple grounds. Asakusa 1 chome, Asakusa 2 chome, Nishi Asakusa 2 chome, and Nishi Asakusa 3 chome.
If possible, do a bit of research ahead of time. Some of the shops in this area date back over 200 years, and are still run by the same families. Find out which ones and search them out. Some of the more famous are: the 'Tokiwado' which sells sponge cake filled with sweet bean paste, 'Bunsendo' which sells fans - known for special orders for Kabuki stars, 'Yonoya' which sells boxwood combs,...
Be sure to:
1) Walk the Hanayashiki Street and check out its amusement park that dates to the mid 1800's.
2) Wander Rokko Broadway. Once quite spectacular, it is still fascinating, even if a little seedy. Along this avenue are traditional theaters, strip shows, a huge horse betting establishment,...
3) Get lost in the covered arcades. Eat in one of the tiny family run restaurants filled with locals.
4) Walk to Kappabashi and check out the plastic model food and stores that sell almost every kitchen item imaginable.
Things I have not yet done, but are suggested:
1) Visit the Asakusa Kannon onsen public bath (sento). It is open from 6:30 - 18:00, and costs 700 yen for adults.
2) Explore the Edo-shitamachi Tradition Museum (not far from the corner of Kokusai dori and Kototoi dori).
3) Enjoy the Western style buffet lunch on the 28th floor of the Belvedere Restaurant, in the Asakusa View Hotel. Views of the area are said to be fantastic.
I also recommend traveling to Asakusa by boat up the Sumida River. For more information, see my previous write-up called 'Tokyo Cruise Boat'.