Listen to the tune enough, or play it enough that it is in your head so that you can sing it completely through from beginning to end. Once it is in your head, pick a starting note and play it slowly without the music thinking the intervals rather than the individual notes. If you don't have the intervals down, this is a great reason to learn them.
As for chord changes, memorize the changes of a few standards like Satin Doll that have commonly used progressions and then try to recognize similar patterns when they appear in the new tunes you are trying to learn. The more tunes you memorize the easier it becomes.