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Is there a place where I can get world records going back as far as they have been measured, and also see when there was a new world record holder?

For example, if the world record for the mile run was 5:00 in 1900, then it was beat in 1905 at 4:58 etc.

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How about Wikipedia?

Check out https://en.wikipedia.org/wiki/List_of_world_records_in_athletics and click (progression) next to the event name which will take you to a page like https://en.wikipedia.org/wiki/Men%27s_200_metres_world_record_progression with a historical progression of the given record.

There was recently a blog post about using this data with R at http://blog.revolutionanalytics.com/2015/12/using-segmented-regression-to-analyse-world-record-running-times.html too.

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