Before doing a workout, even one you normally do but ESPECIALLY before a new workout or stepping your workout up a level, you should do a good warmup session and then some good stretches, targeting especially the areas you will be working. I know this adds time to your workout but it is really worth it.
At the end of your workout, you should do stretches again. These help to ease stiffness. A good hot soak in the bath can help too.
If you are stiff the next day, try walking. This is an excellent exercise for getting all body parts moving again.
You really should warm up in some way before doing stretches. It is not good to stretch cold muscles that have not been doing much. Some brisk walking on the spot, or jumping jacks, or even walk up and down the stairs a few times will help to warm your muscles up, ready for doing stretches. Swimming is a good exercise between workout days, if you don't like walking.