Hyde Park is the traditional first move out of halls. You're 5-10 minutes from campus, on top of every takeaway and bar, and share houses are bigger — 4, 5, 6 and 7 bed groups all fit comfortably. Rents run £140 to £170 pppw — the highest of the shared-house areas, with only the City Centre above. It's louder, busier, more "student", and that's the point.
Headingley is the natural second-year move. Walk time to the University of Leeds is around 20 minutes (or an 8-minute bus). Houses are typically a touch smaller but more recently refurbished — two bathrooms as standard on any 4 bed. Rents run £130 to £160 pppw. You get Waitrose, the Arndale, North Lane cafés and a proper high street feel.
For walk to campus, Hyde Park wins clearly. For house condition, Headingley usually wins. For social life, it depends: Hyde Park has the Brudenell and the Picture House; Headingley has Otley Road and every chain pub going.
Budget-wise, expect Headingley to come in around £10-20 pppw under the equivalent Hyde Park house — Hyde Park carries a premium for being closest to the strip, and only City Centre apartments cost more.
The honest answer for most groups: first year out of halls, Hyde Park; second year in, Headingley or Burley. If you know your group is quiet and studious, skip straight to Headingley or Woodhouse.