Thick wire mesh round your tree trunks immediately! Goats LOVE tree bark, especially if the trees are younger - that's where all the goodies are (nutrient-wise). Also goats are browsers rather than grazers (like sheep), so will always prefer to get the majority of their ration from branches/leaves rather than grass.
Mine get a daily big bunch of branches throughout the summer, and the also love the thin branches of (bare) willow in the autumn/winter - they are kept busy for ages taking of the bark.