No, you do not need an instrument at home to take music lessons, but it does make it easier to practice and improve your skills. Without an instrument at home, you will need to find an instrument to practice between lessons if you hope to improve.
Before investing in any musical instrument, you will want to talk with the music teacher to find the instrument that is right for you. Guitars are a popular instrument for beginners since they are relatively inexpensive to buy. The flute is another simple instrument to learn and is a good choice for students who want to learn to play an instrument quickly. Pianos are a more costly instrument but are a popular choice for families since everyone can enjoy playing the same instrument.