The Gemara in Masechet Eruvin (100) discusses the issue of whether one is permitted to walk on grass on Shabbat, given the possibility that he may uproot blades of grass in the process, unintentionally violating the prohibition of "Tolesh" – uprooting plants on Shabbat.
The Shulchan Aruch (336:3) writes (listen to audio for precise citation) that one may, in fact, walk on grass on Shabbat, because Halacha follows the view of Rabbi Shimon who allows performing an act on Shabbat that might result in an unintentional Melacha (forbidden activity). So long as it is not certain that the Melacha will result from the given action, one may perform that action despite the possibility of a Melacha occurring as a result. Therefore, one may walk on Shabbat over grass of any kind, whether it is moist or dry. One may even walk on grass while barefoot, despite the fact that grass might stick to his feet and thus be detached from the ground. It should be noted, however, that if grass does stick to one's feet, he may not remove it by hand, since the grass is considered Muktzeh (forbidden to be handled on Shabbat). He is allowed to shake the grass off or rub his foot against a surface to remove it, but he may not remove it with his hand.
The Shulchan Aruch adds that although walking on grass is permissible, one who eats on a lawn may not wash his hands over the grass, given the prohibition against watering plants on Shabbat. Although the individual in this case intends merely to wash his hands, and not to water the grass, this is nevertheless forbidden, since washing one's hands over grass inevitably results in the Melacha of watering.
In any event, the Kaf Ha'chayim (by Rabbi Chayim Palachi, rabbi of Izmir, Turkey, 19th century) advises against eating outdoors in gardens on Shabbat, for a number of reasons. Firstly, as discussed, there is the concern that one may pour or spill water over the plants. Secondly, seeds might fall from one's fruits or vegetables and land on the ground, which could potentially violate the prohibition against planting on Shabbat. Finally, gardens very often are not surrounded by an Eruv, and thus carrying in these areas is forbidden. Therefore, it is advisable not to eat in gardens on Shabbat. When one must eat in garden areas, such as on Sukkot, he must ensure not to allow any liquid or seeds to fall onto the ground, and that the area is surrounded by a proper Eruv.
Summary: One may walk on grass on Shabbat, but one may not wash his hands over grass on Shabbat. It is advisable not to eat in gardens on Shabbat; if one must, he should ensure not to allow any liquids or seeds to fall onto the ground as he eats, and that the area is surrounded by a valid Eruv.