Dennistoun is fine, I live there. Don't go south of Duke Street, the 'Drives' (between Duke Street and Alexandra Parade) is nice. Lots of student types, never had any trouble here. I wouldn't go past the rail tracks either.
You don't want to be going north of the M8, Springburn is okish but the surrounding area (Sighthill) is dodgy.
City centre, you'll probably be priced out of though somewhere down High Street will be fine. Perhaps look on Bell Street and Parsonage Square. I've lived there before and it's fine.
Woodlands/Charing Cross is also fine but may be slightly above budget, and Council Tax bands are generally higher over there.
I'd suggest near Strathclyde University (High Street/Merchant City area) or Dennistoun for your budget.
Shawlands is also fine, but it's not walking distance from town. Gotta factor that in for coming home from nights out.
New Gorbals is fine, a lot of money has been put into it recently but it has the stigma with the name.
£600-£700 will get you a nice enough 2 bedroomed place in the West End (Byres Road area) if you want to live out there but why you'd want to be near Glasgow Uni types I do not know!
To reiterate - Merchant City (High/Bell Street) area or Dennistoun is where you want to be.