So, I agree with the other answers, that as we look further away from our Galaxy, we see galaxies as they were in the past. Therefore, by and large, these galaxies are younger than ours. That is, we see them as they were when they had an age that could be at most 13.7 billion (the age of the universe) minus their distance in light years.
Therefore to find the oldest objects in the universe, we must look locally. Probably the most ancient galaxies that we can study, are those that are thought to be the building blocks of larger galaxies. These are the "dwarf spheroidal galaxies", several of which have been identified as companions to the Milky Way and M31, and are probably remnants of a larger population of such objects. Their stellar populations appear to consist of mostly intermediate age stars through to extremely ancient stars.
Other ancients objects include the extremely metal-poor stars of the halo population of our Galaxy and a number of very metal poor globular clusters.
Estimates for the ages of such objects are uncertain, but are likely in the range 12-13 billion years old and so only a short time after the big bang.
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