I think when you are at a point where doing any pushups or just a few is really hard that you can make a lot of progress just doing pushups whenever you can, against a desk or normal or whatever you can do, and working on them specifically. Sure the weight lifting and other stuff is a huge help, but you can also get the ball rolling just working specifically on pushups.
Female, 5'3", 49, Starting weight: 163lbs. Current weight: 135 (more or less).
Starting squat: 45lbs. Current squat: 180 x 2. Current Deadlift: 230 x 2