The approach many people take to planting under trees is to cover the area under the tree with a load of loam, then try to grow perennials in that soil. As logical as that approach may sound, it fails to take into account a fact about how trees grow: tree roots need to "breathe." If you suddenly smother them with soil, you put your tree's health in danger.
"So Tell Me How to Plant Under Trees Without Damaging the Trees"
It's OK to spread small amounts of soil over tree roots, gradually. This gives them time to adjust. But to start your average perennial bed, you'll need more soil than that -- so, for the short term, at least, that's not really a solution to the problem.
In extreme cases (when the area under the tree is dominated by a maze of exposed tree roots), your best option might be to spread a 3-inch layer of mulch over the unsightly exposed tree roots. A layer of fresh mulch will work wonders to spruce up your problem area. To add further color to the mulched area, install container gardens.
In less extreme cases, try planting a ground cover. You must choose your ground cover wisely, however, as some can tolerate the conditions under a tree better than others; for example:
To begin planting under trees with these ground covers, start by applying small amounts of soil over the area, in stages, mixed with soil amendments. When you're ready to plant, carefully dig your planting holes, to avoid harming any tree roots that might be in the way. Then plant your ground covers in the holes. In terms of caring for the plants installed under the trees, afterwards, remember, they are competing with big trees for water, so be sure to water them well. Mulch in between them to minimize water loss and to promote weed control.