I have a Shinamo Tourney front derailleur that doesn't change well from gear 1 to gear 2 (I'm new to bicycles so I don't know standards, so I think I should specify that gear #1 is the smallest). In fact, it does sometimes change, but only after a few minutes of pedaling, and it keeps making a rattling sound as it's still on gear #1, but the "command" is to get to gear #2. The problem only holds when I change from 1 to 2, never viceversa. I think, but I am not 100% sure that the problem might be correlated with my chain falling off a while back (I don't remember the details, as in which gear I was and all that, sorry).
I'm not sure if I got to detail the problem as I should be, just because I'm new to this, but if you are willing to help and feel that I missed some information, please let me know and I will give you the information you need.