Help with planning an extended trip

Ya'll seem pretty smart :), so I wanted to ask if anyone knows what the cost would be, and how to plan for, an extended international trip.

I am currently a college student, and I am wholeheartedly one of those really boring people who has never really done what they want to do in life. I am really trying to turn that around while I am still young, and one of those things is putting my deep desire to travel higher on my priorities list. I have been wanting to visit England for a reeeeaaallly long time, and I don't want to go for some 2-day trip, I want it to be a full summer trip. But there are some serious limitations:

1) though I have full scholarship, money is still very tight for me, like month-to-month tight, and I don't have a bucket-load of support from my parents. (Dad is barely in the picture and Mom is already doing all she can) 

2) I don't want this to be a school thing, I don't mind working during that time though.

Basically, all of that to say that this is something I would love to do in the summer of 2014, between my bachelor's and master's, so that I will, hopefully, have enough time to save up for it.Plus, I really want to go and see if i really am one of those people who would be happier living in another country. I always think of myself that way, but I want to know. 

Is this something that I can actually do, or am I crazy and I should plan to make it a quick week-trip? And if it is something I can possibly finance on my own, what kind of money am I looking at saving? I know that London is crazy expensive, but I don't know enough about England to know where else would be a good place to look at.

If anyone could give me some tips, I would really appreciate it.