Try the following basic exercise, it is not in any specific key. If you know your scales you can apply the same fingering to scalar patterns and go up and down the fretboard in 2.5 octave runs. This excercise helps to get picking accuracy, clarity and improves overall technique.

123
---123
------123
---------123
------------123
etc and

1234
----1234
--------1234
------------1234
----------------1234
etc

Just play the pattern up and down. As it is just a basic pattern you can concentrate on the technique without getting bogged down trying to remember a complex run or riff.

The important things to concentrate on are:
Alternate picking - up down picking for each individual note, plenty of info on this site
Muting - practise dampening the the strings with the heel of your picking hane AND with the fingers of your fingering hand. There is no one technique for anyone, it depends on your own style, length of your fingers, dexterity etc.
Try to pick evenly from note to note and string to string with the above in mind and pretty soon you should be able to eliminate those background noises. Everyone finds this difficult at the beginning but you will get around it soon enough.