I'm not taking Avonex but am on Betaseron, another of the interferon drugs. Usually they'll recommend that you get injected (or inject yourself) right before going to bed, so that you sleep through the worst of the side effects. What I generally did when I started was take two ibuprofen before injecting, and put two and a glass of water at my bedside.
Then inject. If I woke in the middle of the night with fever, chills, feeling awful, I'd glup down the two ibu's and after 20 minutes or so, go back to sleep. Sometimes I'd need a couple more in the morning when I woke up.
The worst of the symptoms subsided for me after the first couple of months (remember with the Beta I inject every other day, not just once a week, so your experiences may be different!) I still take two ibu's before I inject, still inject before I go to bed, and don't usually have anything more the next day.
...I am not a doctor, nor health professional, and don't pretend to be one, here.....