Do I always move my hands straight forward?
Not at all! 'Put' is a directional sign. You should physically move your hands toward the actual location where the object is being placed in your signing space, whether that is to your left, right, up high, or down low.
Can I use this sign for 'putting on' clothes?
No, this sign is specifically for moving and placing physical objects. If you want to say 'put on a shirt' or 'put on shoes,' ASL uses entirely different signs that mimic the action of dressing yourself.
Do I need to use both hands?
Yes, the standard sign for 'put' uses both hands in the teardrop shape. However, if you are holding something in one hand in real life, or if the object being discussed is very small, you might see fluent signers modify it to use just one hand.