Recommendations in Dublin 320K/340K 3 bed House

As a by the by, I think you would be lucky to get a house in Raheny (near Dart and the village )or Kilester for around these prices. I live there and houses tend to get bid up very quickly to 450K plus.
 

Someone has already mentioned Cabra, I would add Dublin 7 (Phibsboro, Smithfield, Stoneybatter etc) area in general. Looking at your list of considerations:
- plenty of public transport between bus & both luas lines (plus nitelinks)
- local schools - can't comment on quality, but transport available
- tesco, lidl well represented - not far from town as above
- family friendly - many former council estates in established areas
- parks - Phoenix park among others - great for cycling, summer events etc
- third level - Grangegorman campus, transport links to city and buses direct to UCD
- hospital - Mater, Rotunda, James's nearby
other - recycling centre, public libraries, cinema, gym/pools, local festivals

On your house size - 3beds in former council estates tend to be 2 medium size rooms, plus a small box room, handy for a WFH office, so just something to keep in mind if that's relevant to you when purchasing.
 
Holywell in Swords still has homes well within that range, schools nearby, 5m walk to big Tesco plus a local Spar. Couple of buses run through it and you'll have a MEtro in a few years. Sister bought round there and absolutely delighted with it. Plus a few of the quieter walks around it.
 
Beaumont is nice and quiet, lots of family there. A few schools round it. Shopping though pretty limited choice. Transport reasonably good though.