Rubidoux: You can develop strength with minimal equipment. Bodyweight training is highly effective and sounds like a good option to begin with for you. Yep, the pushups, pullups, squats, lunges. There are many variations of these movements as well as progressions. I would suggest exploring various YouTube videos for ideas that demo the progressions.
Bands: I like bands. When beginning your training bands may serve the "heavy" part of your training routine. As you progress, you may then opt to use the bands as a warm up to a different type of strength training program. I prefer flat bands from the site below. From this site you may also find lots of ways to use the bands in your training. They are less of an investment than free weights and will get you prepare you physically as well as in a neuromuscular sense for freeweights. The ones I use are the monster mini black. They were perfect for me but I began using then after training with freeweights for about two years. I have used them with deconditioned clients with success. Lateral lunges with them are a little tough though for beginners.
But the idea here is that it's better to do six to eight reps with these things than twenty to thirty...we are working on strength...not endurance.
About the knees....not training the quads/hamstrings is an invitation to injury. It is important to develop and maintain strength in these muscles because they support and just as importantly ABSORB the impact of movement from the knees. Good things are step ups, single leg training, squats...lunges are good but do tend to bother some people with a preexisting condition until greater leg strength and coordination is developed by other methods. Some have greater success with backward rather than forward lunges in this way.
Another minimalist approach to developing strength is yoga. Most people use a sticky mat...highly suggested...but I've been known to go matless in a gym or outdoors when inspired with a little extra energy to burn.
There are numerous great yoga sites but I'll direct you first to YogaJournal.com. You could also get vids from the library to get you started. I found an investment in a beginner series at a proper studio for six weeks gave me a great foundation for learning on my own. Cost me about 100.00 for the series and I was able to learn proper alignment to carry into my own daily practice.