a GORGEous ride
I used to live right in the middle of Corn Street, and one day me and a mate decided to ride out and get hopelessly lost round by Ashton Gate and the left side of the Avon Gorge, as we didn't know the area and we wanted to see what it was like!
this is the exact route we ended up taking! we were pretty lost between Ashton Gate stadium and North Road in Leigh Woods, which is why that portion's such a mess-- but it's a hell of a lot of fun to ride, especially the weird brand-new busway out by Ashton Gate, and you get some beautiful views of Hotwells and the Gorge too.
It's not too tricky a ride! I completed the whole thing on a singlespeed, so any bike with a reasonable amount of gears would be completely fine. I'd recommend a road bike, a lightweight mountain bike, or a sporty hybrid.
NB: the Nightingale Valley segment involves chucking your bike on your shoulder and hiking up the side of the Avon Gorge. foolhardy maybe, but a LOT of fun. it's also easier than climbing up the entirety of Ashton Court estate, which is the only real alternative-- Rownham Hill is a horrible and genuinely quite dangerous climb, and an absolutely terrifying descent. not recommended either way
there's plenty of potential shortcuts! You could cut straight across Clift House Road to the park, for example, or just go straight from the Suspension Bridge back down Park Street (or Constitution Hill and Jacob's Wells if you're feeling brave)- whatever works for you!
there's also plenty of options to go on cycle paths (like the one along coronation road, or the ones across the Downs), or divert down quieter side roads. if you don't want to mix with the traffic heading down Whiteladies off the Downs, I recommend skipping Ladies Mile entirely for either Pembroke Road, or College Road>Lansdown Road>Queens Road. up to you!!