Back in my college days, I tutored a lot. I actually paid my way through college by teaching computer programming to many students. At last count, I think I have taught C, C++, and Java to almost 150 students in total. And along the way, I taught them the basic algorithms they could test out.
Which brings on the question: what algorithms should newbies learn first?
From my experience, I think first-timers ought to start learning simple things (such as finding the maximum of three numbers), then progress towards slightly complex versions of these problems (such as finding the maximum of n numbers). Then they should focus on things such as finding the GCD using Euclid's method; Binary search, etc. Sorting can come next, and I started teaching my students Bubble sort (yes, I know, it's the worst performance algorithm, but easy for people to grasp, compared with Quicksort!!).
Once the students master searching, sorting, max-mins etc., I then taught them recursion. It was quite fun to teach them recursion using the analogy of a staircase ... you go down the stairs, doing something in each step as you go down and then return (or doing something in each step on your way up).
Bernhard Koutschan recently posted a list of the most important algorithms. (thanks to Daniel Lemire for pointing it out). Among my favorites for first-timers, only 2 made that list (Euclid's GCD, binary search). The rest of the algorithms in that list are a bit complex for the newbie CS students to grasp in their first semester ... at least that's what I felt during my tutoring days.
Lemire has also posted a shorter list of the 5 most important algorithms, along with a poll, in his blog. It will be interesting to see what shortlist people come up with.