If you like milk and ginger then you HAVE to try ginger milk pudding (hot or cold). It's a Hong Kong specialty and the science behind it is the most fascinating part (no gelatin or anything). Yee Shun Milk Company is the most famous (with multiple locations), but you can also go to the Australia Dairy Company. They also have other snacks and desserts.
Also a quintessential HK dining experience, cha chaan tengs are somewhere between cafes and diners. People mostly eat there for breakfast, lunch, or afternoon tea. (But who says you can't have HK french toast at night with some wok tossed noodles?) Staples include pineapple buns (bo lo bao), coffee + milk tea (yin yeung), HK style french toast, macaroni or packaged ramen in soup with a fried egg and a meat, soy sauce chow mein, and others (the set meals are the best deal). This list includes the Australia Dairy Company, which was mentioned under ginger milk pudding. An additional from Monique include Cheung Hing Coffee Shop. You can't really go wrong with ducking into any old cha chaan teng.
Hong Kong's "Monster Building" has become one of the most instagrammable spots in the city. Monique actually grew up right near there! Right next to a tram line, the area itself is fun to explore and generally has decent, down-to-earth eateries. It's also near a trailhead for Mount Parker Road, which is a popular hiking trail.
Great for a day trip! You could take the MTR out to Tung Chung and get on the Ngong Ping 360 cable cars, which take you over the South China Sea closer to the Tian Tan Buddha and the Po Lin Monastery (which includes a vegetarian restaurant). You can climb 268 stone steps up to the Big Buddha or view it from below. You could then jump on a bus to Tai O fishing village, where you can eat, buy dried seafood, souvenirs, or take boat tours to see the houses on stilts.
If you like Hong Kong history and/or film director Wong Kar Wai, you may have heard of Chung King Mansions. It's home to many immigrants from South Asia and Africa -- and is known for food from those regions -- and cheap Rolexes and such.
Good for cheap clothes, bags, souvenirs, and other knick knacks! The stalls go for miles and you're free to test your haggling skills. Mongkok is one of the few areas that still has some street food left, so you can wander around after shopping.
More markets! Wan Chai has both a big wet market (for fresh meat, fish, and produce) and an outdoor market for home goods, clothes, and knick knacks.
Also known for street food, dai pai dongs, and cheap goods.
A good spot to get away from the hubbub of Hong Kong. It also has way more than 10,000 Buddha statues.
Lan Kwai Fong is a popular nightlife spot in HK. You can also keep going up the alleys/ long escalators to Soho for restaurants and pubs. (Wanchai also has a good and more laid-back nightlife scene.)
This former colonial police station/jail has been converted into a museum with shops and boutiques. It's also close to Soho or Sheung Wan, which has a lot of hip coffee shops and restaurants, or Man Mo Temple, which is known for its spiral incense.
The tallest peak in Hong Kong -- go on a clear day for some nice views of the city. There's a viewing deck (but they now charge for it) and a nice (and free) walking loop around the hill. The mall itself is nothing special. You can hike up, bus up, mini-bus up, or take the Peak Tram from Central MTR station.