Listen to some cover songs. Find any audio clips you can of musicians covering different types of songs. This will provide you with some ideas about the type of songs to cover, and with the things you can do with different songs. Listen to songs such as the Raconteurs' version of Nancy Sinatra's "Bang Bang" and Johnny Cash's version of the Nine Inch Nails' "Hurt" to get a feel for the different ways you can take covers.
Choose a song to cover. Think about songs you haven't heard covered before, or any song that you would really like to do another version of. Choose your song based on the type of cover you would like to try most. If you like rock music, but will be covering a song by yourself on an acoustic guitar, think about a rock song that would translate well to acoustic playing. Remember, if you have a band, you can make a heavier version of another song. Listen to Bob Dylan's "Maggie's Farm" as performed by Rage Against the Machine for an example.
Learn the song you are intending to cover. Find out the chords to the song to get a feeling for the basic structure and notes of the song. You can either learn the song as it was originally played or learn a simplified version of it. Practice playing it through a few times.
Decide which direction you wish to take the song in. If you are in a band, it is often a good idea to play it as if the song had been written by your band. This helps you establish your own band's sound and make the song your own. If you are playing the song alone, think about what you think would suit the song, or what type of version you would wish to do.
Use the chords or guitar tab of the original song to create a new version. Slow the song down or speed it up if you wish, and remember that you can play chords in any way you wish. For example, if you wanted to do a heavy metal cover of "A Horse with No Name" by America, you could palm mute the opening chords with heavy distortion, and then add in small, in-key fills to rework the track to your style.