What is the best approach to cut carrot into thin strips?
The style of cut is called julienne.  True, a mandoline can make a julienne cut, as well as thin slices.  However, you can also use a knife to achieve a julienne cut.

First, cut the carrot into manageable lengths--2 or 3 inches.  Cut a piece into 1/8 inch slices lengthwise.  Stack several slices on top of one another and cut lengthwise through all layers to make "matchsticks" or julienne-cut carrots.  It works best to have a large slice on the bottom of your stack, not a small one or the round side of the carrot.  A large slice on the bottom is more stable, so you're not as likely to julienne your fingers.  Yes, cutting carrots this way is a lot of trouble, but this is the way to do it with a knife.

There are also other ways to get a similar result without a mandoline.  If you have a food processor, try the large grating disk.  You can also use a salad shooter or other slicer/shredder--there's even an attachment for the ubiquitous Kitchen Aid mixer.  Even a plain old box grater can be used to grate carrots.  Grated carrots may not be as neat and tidy as the ones done commercially, but they're still a good addition to a salad.

Oh, if all else fails, you can usually buy the shredded carrots in the produce section, near the bagged salads.