Great Wall

You should have no probs picking up a tour from your hostel/hotel. Many tours will take you to the Badaling part of the wall. I believe it's the closest part of the wall to Beijing.

If you feel like walking a section of the wall in a slightly quieter location, you could do the walk from Jinshanling-Simitai. (It's not an overly strenuous walk for anyone with reasonably good health. Make sure you have decent walking shoes. And whilst you can buy water and a little food en route, cheaper to buy supplies in Beijing or at Jinghanling before you start walking. There are hawkers along the route if you do need to buy some things but of course the prices are overly inflated.)

There are 3 ways to do this:

1. Catch a bus to Jinshanling, walk the 10-12km to Simitai, and then bus back from there. You'll need to check bus schedules yourself as I didn't do this myself.
2. Cab it. Hire a cab for the day from Beijing. Get a cab to Jinshanling, walk the 10-12km to Simitai, and cab it back to Beijing. It's possible to get a cab for the day from central Beijing for around 800RMB. If there are 3-4 of you, it's a pretty affordable and comfortable option. Get your hostel/hotel to organise this for you and to negotiate the price. Make sure that both you and the driver understand the deal before you depart - i.e. agreed price, agreed departure time, agreed drop-off/pick-up points, waiting time. It's a pretty typical gesture to buy your cab driver some lunch too - or at the least a cool drink. BTW, the 800RMB covers the cost of the cab including highway tolls and petrol. If you wish to take a cab, try to leave as early as possible so you miss morning traffic. 6am is a good time. You will still need to pay for entrance to the wall itself.
3. Join a tour from a hostel/hotel. It should be easy enough to organise this when you get there. "Tours" cost around 230RMB per person from Central Beijing. note that this only covers the TRANSPORT to/from the wall. You will need to additionally pay for entrance to Jinshanling (50RMB) and Simitai (40RMB) sections of the wall.You also have to pay a 5RMB fare to cross the bridge just before Simitai.

The walk should take about 3-4 hours, walking at a fairly easy pace. It's pretty steep in sections so bring proper walking shoes. Sunscreen/hat are also a smart idea. You can buy water in many places along the route (at least once every km, perhaps even more frequently), however the prices are much higher than what you'd pay from a shop anywhere else. So bring your own if you can.

There are touts at the beginning of the Jinshanling section of the walk and they are pretty bloody persistent. They may offer to act as your guide or even try to guide you even if you've said you don't want a guide! Annoying but harmless. One guy who claimed to be a "Mongolian Farmer" followed us for more than 2 hours along the wall, even though we didn't buy anything!!

Food

The Youth Hostel near the Simitai entrance has a great little restaurant with a few veggies options - their MaPo Dofu is so-so, their kale dish is really good and their braised eggplant dish is amazing!