I lived in Easton for 4 years, and it was amazing. It has a much undeserved bad rap. It is full of lovely restaurant (the Thali, Eatern Taste, Cafe Mateya to name a few) and excellent shops I loved it to bits, its so full of culture!
I've also lived in St Andrews its as nice as clifton IMO and not as pricy plus you have Gloucester road on your door step. Montpelier, St Werburgs are also very Nice.
I have lived in Sea mills and it is snoozeville not much happening.
I currently live in Horfield (I get around) and I like it, its near Gloucester Road and the rent and house prices are reasonable.
In short it depends what you want, if you want lively areas that are busy and buzzing I'd suggest Clifton (if you can afford it), Easton, Stokes Croft and Montpelier maybe even st george.
The Boho areas are Montpelier, St Werburgs, Whitehall, and Easton
If you want family areas try St Andrews, Southville, Bishopsworth, Henleaze, Downend, Totterdown, Redland.
If you want directly near the city centre try Hotwells, St Pauls (I've worked in St Paul and in my opinion it gets a bad rap as well), KingsDown and Stokes Croft.
If you want BIG houses and have loads of space cash try Sneyd park, Leigh woods, Long Ashton or Stoke Bishop (Clifton as well)
I would personally avoid Knowle West (some parts of Knowle are fine), Withywood, Hartcliff, Southmead, Lawrence Weston, henbury and Shirehampton there is a lot of relative poverty in these areas and they are quite far from the city centre.
Bedminster, Bradley stoke, Bishopston, St Anns, fishponds and Lockleaze are meh... I can't say too many bad thing about Lockleaze as my wife was born and raised there.