This is a common vocal warm-up exercise that doesn't feel like an exercise. Lip trills are done by putting your lips lightly together, breathing out and making enough sound so that your lips flutter. You can make any kind of sound at any volume or use scales to warm up. Tongue trills are done by putting the tip of your tongue on the roof of your mouth, breathing out and making sound at the same time.
Scales are a necessary exercise for any singer or musician. It's a strengthening exercise that also promotes an understanding of music theory. It also helps with intonation. If you have a keyboard or piano, practice your vocal scales while playing them. Otherwise, you can purchase CDs of music scales to practice to. For a vocalist, practicing scales will also help with ear training, which is helpful in working on harmonies.
One of the best vocal lessons is to sing songs that are good for your voice. Working on scales and exercises will help to strengthen your voice and improve your intonation, but working on entire songs will help you identify the weak parts of your voice and what you need to work on the most. Record yourself and listen to the recording. Make notes on the parts that need work so that you can strengthen those areas of your voice.
There is a wide variety of singing programs available for purchase that enable you to vocalize at home. Most programs include breathing and warm-up exercises and instruction on different parts of your voice and how to take care of your vocal cords. Some will include also exercises on developing a belting voice and falsetto, as well as pitch and vocal techniques. You can learn these lessons at a fraction of the price that a private teacher would charge, and you have the convenience of practicing as long as you want when you're at home.