A quick brain dump:
Johansen - as mentioned, decent bike path from Steese Expy to University. Not very well used most of the time, reasonable caution required, but you can usally move safely at speed.
College Road: worthless, sidewalks only, probably the main road I get harassed if I ride on the road (which really is the only option there).
Mitchell: I just ride the shoulder when I (rarely) need to ride on that road, but don't know if it's legal or not. It's plenty wide and I consider it safe.
In town, the only real drag is Airport Road because you're not supposed to ride on it and it only has true sidewalks, nothing close to being a decent bike route. You can pick your way along the access roads on the north side for a lot of it.
Parks: decent bike path to Ester, rideable shoulder all the way to the Pedro Monument and beyond to Nenana.
Gold Hill Road and Old Nenana Highways are nice rides and can loop w/ the Parks.
Farmer's loop - decent bike path the whole way, plus the shoulder is nice and wide and very ridable (sp?). Bike path is really multi use with kids and walkers, so the shoulder is often much better option if you want to ride at speed.
Steese Expy north: I usually ride the shoulder from Airport or the Johansen to CHSR and beyond, but you can avoid it by taking the bike path north from the Johansen bike path, up to Farmer's Loop where you can get on the old Steese, and get actually all the way up to Goldstream and Fox on the Old Steese. It's a very pleasant ride on the Old Steese after CHSR, just "OK" before that (no shoulder, cars driving close to same speed as expressway - which is why I often just ride the Expressway shoulder between town and CHSR).
Goldstream - as already described. Very nice ride, low traffic, but a lot of chip seal and narrow shoulders.
Loop various portions of Goldstream, Old Steese, Farmer's Loop, Ballaine, Johansen, Miller Hill, McGrath... many options which you should be able to figure out with a map.
Steese and Eliot Highways out of town, north and east of Goldstream Rd: excellent riding, hilly, reasonable low traffic.
CHSR: good wide shoulders out past Nordale Rd, then narrower shoulders on and off the rest of the way, but nice riding and many out and back destinations with a break alongside the river. Check the FCC schedule - there's an organized ride to CHS every summer that's a lot of fun.
There's a lot of other loop rides, too many to mention, but look at your map and link up various sections of Cripple Creek, Isberg, Chena Ridge/Chena Pump, McGrath, Skyline, Sky Ridge, Summit along with the above major roads, etc. etc. Are all very nice rides and you can make loops depending on how far you want to go - lots of places to start on all those loops.