I have some tables within one single story (not in separate text frames). While I can easily change the width of the columns, I fail to move some of the tables from their left margin (of the type area). It's no problem with the right margin, though. – Am I missing something?
Tables are contained inside a paragraph and obey the paragraph alignment and inset options you have set. In other words, if the table is in a paragraph set to left align, it will hug the left margin or any left inset applied, but it can grow beyond the right margin. If you want it to expand beyond both margins, it must be in a center-aligned paragraph.
You can see the whole table like a paragraph: place the cursor in front of the table, you will see a cursor the size of the table. Then simply apply a left indent!