If you have no Pokemon above 60, it would be wise to do shopping first. I suggest buying 3 to 6 revives and 5 to 10 Hyper Potions. It will come in handy. You will be reimbursed big time when you beat them.
Know your opponents. Know their Pokemon they use and what is good against them. It is usually good to know what they use first and what move. For instance, Lance uses Gyarados first, and he always does Rain Dance (I use the Silver version, so if you have Gold, it might be different, I have no idea!). If you usually go first of the Pokemon, you know to do a high powered move. If you have a weaker Pokemon, on the second move, you could do a self-healing move or something.
Heal when you need to! Since mainly you don't have all six of your Pokemon above 40 or 50 when you reach the Elite Four at first. So, when you are almost faint or already fainted, switch to a weak Pokemon you know can't do squat against the Elite Four, and put a revive or hyper potion on them.
Use a great type-difference Pokemon! Fire against water, Water against fire and rock, etc., etc., etc.! This really helps.
Use your breaks! If your Pokemon are looking pretty low after a trainer, heal them, or whatever. And remember to save.