Vinzer Deling's Train

In the train, you will first battle Deling's double, and then after wining this easy battle (just use physical attacks and Guardian Forces), you will have to fight a monster that uses a lot of status attacks.  Luckily, you can draw Esunas (these cure all status attacks) from him, so junction the Draw command to each one of your party, and as soon as someone is affected by a status, draw and cast an Esuna on that person.  This monster is an undead monster, so using Recovery magic on it will harm it.  Guardian Forces also work quite well.  Just keep an eye on your HP, and draw and cast Esunas, and the battle won't be a problem.

Shumi Tribe stones

The water stone is at the artisan's house, he uses it for washing, the blue stone is behind the Laguna statue.  The shadow stone is in the shadows near the Ultima draw point and the life stone is in the tree roots of the tree sprawled over a rock.  I think that's all of them, anyway!  ^.^

Winhill Vase Pieces

The first vase piece is in the statue of a knight.  Go inspect the statue and it will topple over and a vase piece will appear.  The next piece is in the flower shop.  Go look at the white flowers and talk to the florist, inspect the white flowers again and the vase piece will appear.  The next piece is found when walking across the path in the fields where a chocobo is crossing the field.  Run into the chocobo and it will drop some items, including a vase piece.  The final vase piece is in Raine's house.  Inspect the white flowers, then go talk to the cat.  It will turn into Raine's spirit then the vase piece will drop from the sky. Please note that I haven't played the game for months, so these may be incorrect.