What supporting muscle groups need to be strengthened?
All of them! Cycling is a full body sport, despite what some people will say. Ever ridden a long ride on a road bike? You may have soreness all the way from your neck to your toes. Some of this can be from poor bike setup, but that's another topic entirely.
The major muscles are obviously going to be legs, back, and core as these are the muscle groups that support the motor (making the bike go). Next up is your upper chest and back, shoulders, and arms which all support the driving (telling the bike where to go). If you ride on bumpy roads or trails you know how much your arms get beat up!
In what order of priority should those muscle groups be addressed?
Do not train exclusively one set of muscles over another, that's just generally bad practice. You should look to strengthen all your muscle groups. Now, if you have a specific objective in mind, like "I want to climb hills like a mountain goat." you may want to start by targeting your weak points (leg power) or strengthening the needed portion (hamstrings, quads, calves). For all around cycling fitness, don't just work the "mirror muscles" aka the big ones that have lots of definition. Supporting muscles are what keep you from getting injuries in your joints or excessive stresses in your major muscles, so don't neglect those.
What is the optimal strength to maintain for each of the groups? By optimal strength I mean how much effort (10min 3x wk for example) do they need? If it depends on how much biking then please answer for a nice spectrum of distances such as 60mi/wk, 150 mi/wk, 300 mi/wk.
Again, this depends on your goal. If you're a weekend warrior looking to do some nice leisurely rides around the lake or through town, you'd probably be OK working out 2-3x per week for 30 mins doing basic bodyweight strength training. But if you're wanting to prepare for a bike race, triathlon, cross-country epic, you'll want to do 3-4x (or more) for 45-60 minutes a week to not only maintain, but develop and improve your strength.
Cross training (running, swimming, lifting, etc.) can be very valuable to any athlete from amateur to pro. Look at various programs and find what works for you. Body types, goals, and external limits can help you plan the type of program that will work best for you.
*Note: I'm not a certified/licensed trainer/doctor/physician but I've been doing sports since ever of all kinds, and currently race amateur class mountain bikes.