When you are training, your whole body adapts to the stress you put on it. The speed, how different structures of your body adapt, differs. Muscles and tendons will adapt faster than joints and bones. Under normal conditions you don't need to explicitly train your tendons – and I doubt it is possible at all – they will develop as your muscles grow, and so will all the other connective tissue.
Whether static stretching is good or bad is a highly disputed topic. I personally don't stretch statically as it increases pain tolerance. As cool as pain tolerance might sound, pain is a warning signal, and I don't want to train this away. Some people think static stretching might be useful after exercising; I personally am not really convinced, and too lazy to do it anyway.
You should warm up before every workout, you can use dynamic stretching. The main difference between static and dynamic stretches is that you only bring your muscles and tendons in the fully stretched state for a short period, but repeatedly. The movement gets your blood flowing and readies your whole body for the exercises.
Eating the right stuff to support your connective tissue? – I don't think that this is necessary. But anyway, a big portion of your tendons consists of Collagen and the body needs Vitamin C to actually put it to use. I wouldn't supplement that, just eat an apple.