I want to use RasPi to display continously some simple data (like today's weather). The thing is that I don't want to use this screen to display the Raspian console on it. I would like to control the screen directly from GPIO pins and construct the basic visual information that I need. Do you have any suggestions? Could an old phone screen work?
Your question is very broad but a 16x2 I2C display is probably what you are looking for. It only takes four pins because it's I2C and you can fit 32 characters on it which is enough for most small applications. You can get bigger sizes also.
There are libaries for python and C to drive these. Not to mention they are very cheap.