Why does my multimeter always show a value of 1 for resistance?
The display with the 1 in the left most digit is normal.  

It indicates that the connected resistance is too high to measure on the selected scale.

If you plug in the leads but don't connect them to anything then the resistance is (as far as the meter is concerned) pretty much infinite, which it shows with the 1 in the left most digit.

Connect the leads to the proper jacks (sounds like you have, red to the Volts and Ohms jack and black to the Common) and short the ends of the probes together.  You should get a value very near zero.

If you don't get that zero reading then post a picture of your meter as you short the leads together showing the meter (and its settings,) the connections of the leads and how you shorted them.  You may be doing something wrong.  If everything looks good, you may have a bad lead or a meter with a bad jack.  More likely, though, is that you are doing something simple wrong.



From the answer you've now posted, it appears you were doing something simple the wrong way.  An incorrectly connected lead is an operator error.