Beijing and the Great Wall of China are two places all first-time visitors should put on their itinerary. To avoid tourist scams (and in Beijing there are many) read about it in your travel guidebook before you go to Beijing.
I love Beijing! The air quality is terrible, the traffic is a nightmare, and all of the tourist sights are crowded, but I still love Beijing because there is so much to see and do. Locals are very friendly, but make sure you bring a Mandarin phrase book. I easily spent six days here.
Don't stay near the Tian'anmen Square area. It's full of tourists traps and pickpockets. Everybody tries to rip you off. Noisy everywhere. But Tian'anmen Square and Forbidden City are great!
Great city, loads to see, forget about Mao's tomb, 3 odd hours to queue and no backpacks / water bottles allowed. The Wall is amazing, but Ming tomb's a waste of time. Tianeman Square, the Forbidden City, the Buddist temples, and the parks are great and the locals are very friendly. A major problem is language, you must have English - Mandarin cards or else... A funny story is an American tourist who used the box of matches from the hotel. The taxi took him to the match factory! Otherwise great. Train travel is very easy - use SEAT61.com, a great website about train travel.
Beijing is a massive city with a lot to do for the traveler. You obviously have to visit all the traditional tourist attractions (the Great Wall, Forbidden City, Summer Palace, Tiananmen Square, etc.), but there are thousands more things to do based on your interests. You can immerse yourself in the Chinese life by walking around different areas, you can stop and eat at small cheap restaurants, go to parks... the subway system is very good, cheap, clean and safe. So pick Beijing hostel close to a station, you will have to walk quite a bit anyway... but at least if you're close to the stations that'll help.
Beijing hostel staff are also usually very friendly and eager to help, so they will be your best resource to find ways to enjoy Beijing. They can write a few Chinese characters for you, tell you where to taste the best tea, help you fix a bike if you use one, make copies of your passport for your next visa, and they can tell you where you can eat dog (!) or the more traditional Peking duck for cheap, and how to go clubbing safely. All Beijing hostels will also have organized tours to book (try to chose the ones which don't include shopping, even if they cost a bit more because being forced to stay in jade or silk stores for hours instead of enjoying the Great Wall like you expected is not the greatest way to spend a day in Beijing).
Beijing hostels in particular and hostels in China in general usually don't have a shared kitchen for their guests to use. However, they generally do provide fast-food sorts of meals (which are usually much more expensive than food you can get from a nearby restaurant). Instead of eating the pizzas and BLTs at your Beijing hostel, be adventurous and try out the local specialties. We recommend that you plan to stay at least a week if you want to truly enjoy seeing Beijing.