Interestingly, a smaller tire is slower at the same cadence in the same gear. This is caused by the reduced rolling diameter reducing the overall drive ratio. However, assuming you aren't already spinning out your highest gear, you should be able to push a higher gear than you would have before. The biggest change when going to a smaller tire is that it is much lighter and therefore easier to accelerate. The reduced rotating mass will also help to make the handling of the bike more responsive, as it will take less user input to overcome the centrifugal force created by the spinning tire (this also makes the bike a bit less stable; for example riding with no hands, the bike will feel a bit twitchier).
Yes, I would recommend buying tubes that match (and remember 26x1.25" is not the same as 26x1-1/4"...not in the world of bicycle tires anyway)