Go to Lin Heung Kui on Des Vouex Rd West for dim sum. They do the cart experience, it's cheap, and the dim sum is really good. It was one of the best dim sum places that I found there. They don't speak any English though but they servers got to know me when I started going every week. All you need to do though is tell them what tea you want, how many in your party, and then you can just point at the stuff on the carts. They have Char Siu Boaloa Bau which is awesome but always runs out quickly. Meal typically costs $40HK and their Pu-er tea is not too shabby (Bo-ley). If you want some good tea I know a good tea house in Sheung Wan. Run by a family from Chui Chow, real nice guy.
Sham Tseng is a good place to go for roast goose. You can catch a bus near Des Vouex in Shek Tong Tsui to go out there if I recall correctly. I can't remember if I liked Yue Kee or Chan Kee better. (Expect to pay around $100-200HK for your meal). Sha Tin has good places for roast pigeon too.
One of my favorite restaurants in the Cantonese style is Wing Wah. They have Wing Wah in Wan Chai and Dai Wing Wah in Yuen Long. Obviously the Wan Chai branch is much more accessible but be able to read the chinese menu, the English menu (if they deign to give you one) is limited. Again this is around $100-200 a meal. Bring friends.
Under Bridge Spicy Crab is a famous joint though I think the crab was getting a bit overpriced. But the good thing is that the non-crab dishes are very good and were reasonably priced. Some even came with that wonderful fried garlic that the crab comes with. You can find them out on Jaffe between Wan Chai and Causeway Bay.
There's a joint called Ba Yi on Water Street in Sheung Wan that's good. They specialize in lamb dishes but also give the camel a try, my favorite dish there. There are a few sushi places in Kennedy Town along Belcher's St. that are decent.
In general though, look at openrice.com for restaurant reviews. Their English part of the website expanded rapidly since 2008.
Things to do are hiking (there are four main trails: Wilson Trail, Hong Kong Trail, MacLehose Trail, and Lantau Trail). The trails on Hong Kong Island are nice, particularly the one that goes from Stanley up through the Wan Chai Gap and the trail around the Pok Fu Lam Reservoir. I also like the trails around Shek O (there isn't much shade on that one though), Shing Mun, Sai Kung, and Plover Cove. The Gin Drinker's Line trail along the range that separates Kowloon from the New Territories has some abandoned pillboxes and trenches from WW2. I should have thought to lend you my hiking trail books if you were interested. If you do go hiking, bring lots of water and sun screen for this time of year. I usually went through at least 3-5 liters of water in a day of hiking.
You can also check out the shops in Mong Kok, the flea market in Sham Shui Po, the Golden Computer Center in Sham Shui Po. You can take a ferry for a day trip out to Macau (casinos, a lot of interesting colonial architecture if you're not interested in the tables). You can also take ferries to the outer islands for a little hiking and seafood dinners (like Lamma Island). The Big Buddha in Lantau Island is a bit touristy though but it wasn't bad of a trip. Long Kwui Fong (LKF) is the center for cheap bars and clubs for the westerners. I think it's overrated but that area in Central has some nice Western restaurants, so does Ashley Road in TST.
There are the racecourses in Happy Valley and up near Sha Tin. When the races are not on, Happy Valley is open for joggers (they have showers and lockers there too).
The Peak is probably the one that you've heard about. You can hike up there from a trail that starts behind the Chow Yee Ching building in HKU. You can do that just one evening as it only takes 45 minutes of vigorous walking to get up to the Peak. That trail also branches out to the trails that run down to Pok Fu Lam reservoir, Aberdeen and Stanley Bay if you have a whole day.
There are various museums. Not too big but they're cheap and you can spend an afternoon there. There is the HK cultural museum in Sha Tin, there's the Coastal Defense museum out in Shau Kei Wan for example.
There are also several temples that you can visit. Man Mo temple is a small one but accessible in Sheung Wan. Out in Sha Tin is the 10,000 Buddha monastery (you would probably want to combine that into a general trip of Sha Tin).
In the Admiralty you will find the Botanical Gardens and Hong Kong Park (which has a Teaware Museum if you so fancy). Victoria Park in Wan Chai is more for sporting. People go there for jogging, football, tennis, basketball, etc. There is also a small pond for RC boats. The main Hong Kong library building is across the street from Victoria Park too but they have branches all over the city. Their English library is a bit spotty, I had to travel to different branches for specific books.
That's about all that I can think of off the top of my head.