1. From Lucy Cavendish to King’s Parade
This is the easiest route I can think of. The whole way, you have at least a designated lane, and Grange Road is pretty quiet. The hardest part is from Lucy Cavendish to where the shared pavement starts just round the corner; I can think of no good alternative but to walk. This way is also a relatively long detour, but very easy to cycle down. If you want, I can give more information on routes, e.g. to Homerton, from here.
You can cycle on the pavement for most of Madingley Road; unfortunately, not the bit on the side near Lucy Cav if you turn right onto Madingley Road on there. If you can cross that road, you can cycle on the pavement on the opposite side, though it is narrow. I do not actually recommend this; I’d walk the bike round the corner to the right, and cycle after the pedestrian-only bit.
Grange Road is really nice and easy to cycle down, with designated cycle lanes on either side for most of it. Just be aware that if a Uni 4 bus is in front of you, it will likely stop in a bus stop in the cycle lane
The left turn at the first crossroads on Grange Road, coming from Madingley Road side, takes you on a cycle path/foot path past the University Library to a crossing on Queens’ Road; crossing there and continuing gets you to ‘Orgasm Bridge’, by Trinity Hall. (Garret Hostel Lane). Turn right out of there (it’s a really quiet back road) then left down Senate House Passage to reach King’s Parade by quite a quiet route
Slightly harder variation
Between what Sarah said and the previous route: pretty much the same, but with the additional difficulty of a right turn on a roundabout. I find it easier than a right turn at crossroads, but still a little stressful so tend to avoid this one unless I feel really confident on the day. Queen’s Road doesn’t have a dedicated cycle lane, and it is quite a fast road, but it is wide and doesn’t have too strong a camber. From Queen’s Road, you can turn left onto Garret Hostel Lane as in the easiest route, or left at the crossroads at the end onto Silver Street to reach town.
Via Castle Hill
Alternatively, you can head up via the back roads onto Castle Hill, where it’s a not-that-difficult right turn to go straight across the crossroads. I don’t think the roads from Lucy before Castle Hill have a dedicated cycle lane, but they are really quiet so it’s no problem, and Castle Hill has a dedicated cycle lane, with cycle box at the traffic lights to make it easy to go straight across. This also requires a right turn into Trinity Street where Bridge Street becomes one-way in the opposite direction, but it’s generally fine. If you get unlucky, you have to wait for a taxi or two.
I do not recommend turning right at the crossroads at the bottom of the hill ever, though.