The order may not make any difference. You should use whichever makes you feel more comfortable. In general, heat will increase blood circulation to the sprain or bruise which helps with the healing process. Heat will also decrease pain by relaxing the muscles around the injury. Ice reduces swelling and lessens pain by numbing the area with cold.
Ice also increases blood flow after it is removed from the injury.
It is very easy to make your own ice pack. You can use ice cubes from the freezer and place the ice cubes in a bag. You can also freeze water in a styrofoam cup. After it has frozen, you peel the styrofoam off the ice. It is also possible to buy reusable ice packs at the drug store.