Weather around that time is T-shirt warm, although if it rains it can get a little chilly but for the most it should be pretty warm.

Depending on how many people you will be travelling with you might be interested in staying at a hostel. Often they offer day trips to the sights around Beijing for a reasonable price, with guides, and an opportunity to meet other travelers who are going to a lot of the same places you'll probably be going. I don't think any one place is more safe than the other, there are dark hutong alleys all over the place that are no more scary than the busy streets in broad daylight. Hotel or hostel, choose something in the Tiananmen area and you'll be withing walking distance to a lot of places.

You can buy a transit card at any subway station, for a 20 yuan deposit, and you can put as much on as you'd like. Subway is 2 yuan per ride, and the buses are 1 yuan but it depends how long the bus route is if it will be more or less. I'd advise about 20-30 yuan for about a week. Check out www.bjbus.com

English signage and speakers can be found all over the place, post Olympics, so it shouldn't be hard to travel through the city with a travel map showing english and chinese characters (they're not hard to find). That way if you need to take a taxi or ask someone you can point to the map area.

Other than that, have the name of the hotel you're staying out printed out in Chinese and English before you arrive, collect business cards to places you like and eat at restaurants with picture menus!

Happy travels!