You want to provide more details and information. In short all we can do right now is say "When the two objects are touching have them move together". You might also want to take weight into account and slow down the movements to simulate the first object struggling to move the second.
So say A is pushing B. If A is coming from the left, upon collision it will push B to the right.
If A is coming from above, it will push B downwards.
This quickly becomes pretty complex if you have multiple objects, where you can push an object which pushes another object and so forth. You're verging into full physics library territory here, so I would recommend you look into using an existing 2D physics library such as Box2D or Squirrel. Hooking up to these libraries is very easy, and they provide decently robust physics under a free license.
My game is not going to support multiple objects pushing each other. Integrating my game with a 2D physics library is just to much work for such small game. It would be like putting a 500 horse power engine in a piece of junk from the seventies.