Why would there be occasional hesitation with acceleration?

There have been a few problems with this vehicle, but the most recent is after obtaining a new battery the vehicle's lights and the sound from the radio, along with the speedometer would "pulse" at certain times. Not any particular thing would instigate it, or make it stop. Then after a few months it started having issues with acceleration usually after a particularly bad case of the "pulsing". It would do it once, or 10 times, then eventually "catch" and drive normally. I thought it seemed like an oxygen sensor, however it seems to be closely related to the "pulsing" so now I don't know what it could be.