Travelling from Beijing to Great Wall Of China

As you said, you have limited budget and time so I suggest that you can take the subway to scenic spots for sightseeing. Beijing subway is very developed and convenient, and there are many lines that can go to various scenic spots. It can save time in traffic jams and much money. You can go to the corresponding places to buy Beijing transportation"One Card Pass" , and also can go to the designated subway stations or in some newsstands.

If you go to forbidden city, you can take the subway line 1 and get off at Tiananmen East Station. If you want to climb great wall, you have to take the long-distance bus because all great walls are far away from the downtown. Taking bus is the cheapest way but takes more time.