Zola is right about Southville being more gentrified than Bedminster. Parts of Bedminster have 'blurred' into Southville. Streets around Chessel St - Pearl St, Ruby St, Aubrey Rd etc are very nice (estate agents will call this The Chessels). It's possible you were looking at Bedminster Down, which although is not an utter dive - is not what I would class as very 'mc'. The houses in Southville are bigger Victorian terrace houses with small gardens. Many Bedminster houses are smaller with backyards.
Historically the gentry lived in North Bristol (Clifton, Redland etc) and the 'workers' (shipworkers, minors, tobacco factory workers) lived in the smaller terraces South of the river. There is a small left over snobbery about this. However the now elected Bristol Mayor George Ferguson helped to regenerate the area around the old Tobacco Factory (North St). It benefitted from the property boom of the early 2000's where young professionals from Clifton etc realised they could get a large family home near to the city centre relatively cheaply. North St now has bars, 2 theaters, restaurants, independent shops, bakeries and a Deli etc. It will never have the grandeur of Clifton but its got a nice community.
My Dh moved to Bristol 25 years and has lived all over. He said if money was no object he would choose a large Victorian Villa in Redland. I've only ever lived in the South.
Have a look at Long Ashton. I know plenty of people there. A more village feel but a spit throw from Bristol. Traffic is v congested in the morning though.
Re schools - Bristol has no grammar school system so historically anyone that could would go private. Therefore the state schools declined badly. They used to languish at the very bottom of the tables. However, they are getting much better and the percentage of parents going out of county or private is declining slightly (according to last years local paper!). We've yet to decide on senior school yet, it's very possible I'll go local - possibly Ashton Park or Chew Valley. However, many people I know still send into Bath and N Somerset. It's not unusual.
On the whole I love living in Bristol. It's a great city. Yes, you can drive easily from North to South .....rush hour depending.
Let us know how you get on x