Hi,
Regarding 'dodgy' areas. Well obviously that depends on what you would call dodgy, but I think it's fair to say that the center of town is nicer after you get north of the railway line / canal. However, some of the nicest (and most expensive) streets are relatively close to the line, e.g. Leam Terracce and Russell Terrace, both being south of the river but north of the train line and canal. Going out of town north you've got Lillington and Cubbington to the north which are generally nice areas. Again I would say nicer than south leam - Sydenham or Whitnash say. Having said that there are some modern modern red brick estates (of the constructed from platerboard variety) down south which don't have the 'rough' look some spots have - they just look like red brick estates everywhere else!
Having looked on rightmove.co.uk a few times myself recently it's definitely true to say that the properties are cheaper south of the train line / canal. Some streets though seem to be unreasonably cheaper, e.g.
Leicester Street, Hill Street, Norfolk Street.
Bear in mind it's a fairly small town, so I don't know how you'll adjust coming from london. Warwick is very close, Coventry and Birmingham are close enough for a quick shopping trip or night out in a larger city. Schools I don't know about.
You migth want to consider living in Warwick or somewhere in between (the two towns sort of merge into each other anyway).
Myton Roadis a nice (and probably expensive) road with lots of family homes and a big school.
locster