Doctor Who...I really want to watch that series. Would it be alright if I jumped into the current season, or would I have to watch a lot of the older seasons of the show to have any semblance of what's going on?

Jumping right into this season would be possible, but perhaps not the best option. There are two relatively recent "jumping on" points that wouldn't require too much time investment.

The 11th Doctor era began with Series 5. With Series 7 taking a break very shortly, you could easily catch up before things resume in 2013.

If you wanted to completely bring yourself up to speed with the modern Who era, you could always go back to Series 1 and the 9th Doctor era. There are occasional references to events from the 9th and 10th Doctor eras in the current stories, so if you have the time, it's not a bad idea.

But IMHO, for absolutely new folks, the best place to start is Series 5, and then backtrack as time allows. (And there's really no need to go back to the enormous amount of Classic Who material unless/until you've absorbed the Modern Who episodes and decide you need more!)