Allestree/Darley Abbey is top of the ladder.
Littleover, Mickleover, Spondon, Oakwood, Chellaston areas are decent for the most part.
Chaddesden, Alvaston and Mackworth have some alright bits but also some rough bits.
Allenton and Normanton generally rough.
Going by what you request above, I’d say look at Nottingham Road area of Chaddesden, Spondon or Alvaston areas. Chellaston’s good for M1 south access.
Derby’s a pretty decent city for cycling in, you can cycle from the edge of town to the centre/train station in 25-30mins. There’s a converted canal path that runs from Chellaston right to the station and all traffic free.
I lived in Oakwood for 4 years until recently. Nice enough though characterless…2 bed semi with decent garden and driveway for £475 a month (though our rent never went up during the 4 years so possibly close to your budget now). Cycled to work in the town centre in 15 mins and some local shops nearby.
Riding wise – good bridleway network to the south of Derby (there’s a regular fast paced XC Monday night ride in that area), lack of network to the north but still some decent cheeky trails.