With Cornmarket Street closed to cyclists, Sustrans route 5 provides the easiest and safest north-south route, skirting the centre along New Inn Hall Street.
You can’t dodge the climb of Headington Hill, but you can avoid the busy road. Oxford University has opened up a permissive route across the University Parks; combined with recent improvements to Jack Straw’s Lane, it makes a lightly-trafficked quiet route to the city centre. The route can be impassable when the Cherwell is in flood: check with this useful page.
Getting across the city centre without hitting a pedestrianised street takes experience. From east to west, follow Sustrans routes 51/57 via Broad Street. The highlight is New College Lane, a delightful hidden lane round the back of colleges. Take care at Magdalen Bridge and the Plain, where all the roads from East Oxford converge.