Start by drinking half a gallon of water a day (yes, that's right). Singers have to be super-hydrated. The water thins the mucus that is on your vocal chords. Once you start drinking this much water, you will feel more vocally agile. It's hard to drink this much all at once so when you first start out, keep a water bottle with you and take a sip every time you think about it.
Prepare to practice: Loosen up your neck and shoulders by rolling your head in a circle a few times. Loosen your jaw and lips by pretending to chew gum and exaggerate your lip movements while saying "look" and "loud" repeating them quickly (look, loud, look, loud) until you feel loosened up.
Find some vocal exercises online. There are many free resources on Youtube.com, search for "vocal warm up" or "vocal exercises." You want to focus on the warm ups targeting tone, vibrato and belting. (Do not belt until you have practiced it enough and can do it without feeling any strain.) You can also find singing books with vocal warm up cd's for under 15 dollars.
Practice these exercises a few times. They will sound very funny and you may laugh until you get used to them. These exercises do sound funny and you may not believe that great singers have to make all of these funny sounds, but they do. (Note: You may want to do this when no one is home of practice in your car until you get used to people hearing you practice.) Try to practice these for at least 15 minutes a day (30 is optimal.)
After you are done your daily warm-ups, take out your sheet music or lyric sheet. Lyric sheets can be found by searching for "Song title" + "Lyrics" in a search engine. Print the sheet out.
Practice singing the song. You can sing a long with the recording or without music. Never sing while listening to the music in earphones, you will not be able to hear yourself. After practicing a few times, record yourself singing.
Listen to the recording. You may be mortified. Do you really sound like that? Yes and no, it does not matter how bad it sounds right now, what you want to do is mark on your paper at what parts you sound bad or had problems singing. Practice the song again, focusing on the parts you do not sound good singing.
Repeat step 7 until you have smoothed over most of your trouble areas.
Sing sing sing, the more you practice, the better you will be. If you practice for 15 minutes each day, your voice will grow stronger and your tone will be brighter. Remember, your vocal chords are muscles, you have to work them out for them to grow stronger.