Fruits, vegetables and grains are basically always kosher, but they must be insect
free.
Wine or grape juice, however, must be certified kosher. (Because wine was used in the sacred service in the Holy Temple -- and because it can be defiled through its use in pagan rites -- Torah law requires that only wine produced by Torah-observant Jews be used.)
Produce grown in the Land of Israel has special kashrut requirements. "Tithes"
must be separated before it can be eaten, fruit from the first three years
following a tree's planting ("orlah") is unfit for consumption, and special laws govern the produce of shemittah, the sabbatical year.
For other issues that can possibly affect the kashrut of vegetable products see Wine, Bread and Cooked Foods and The 1/60th Rule.