hey YuriTokoro,
the difference between the words 'legen' and 'setzen' is that they are two different vocabulary. 'legen' means 'to lay something down' and 'setzen' means 'to sit' or 'to place' something.
So for example you can lay a puppet onto a table (then it is lying on the table [horizontal]). But when you sit or place a puppet onto a chair then it is sitting on the chair. But of course you can take the verb 'to put' as well...^^
I think just the first sentence is right or natural but the article before 'Tisch' has to be changed. For the second one there is no possibilty to take the verb 'setzen' for dishes. So in some way it depends what kind of thing you want to move.
1. Ich lege die Teller auf den Tisch. (you wrote 'dem Tisch')
I hope I could help you in some way...
