Yes you can say "I want to plant a plant" but they will not know what plant you will be planting. It is weird to say it so i wouldn't recommend it. "I want to plant a vegetable"

"to plant" is a transitive verb so yes, you can say "I want to plant [object]" and it's syntactically correct to say "plant a plant" but you might want to use another object to specify what you want to plant. "I want to plant [a bush/tree/sapling/flowers, etc]"

There are many horticultural and gardening terms for specific planting techniques. Genericaly there is a chinese saying ´Those who plant a garden, plant happiness.´ Seeds are always sown. Potatoes and onions are lined out in the soil. Young vegetable plants, normally grown in pots first, can be transplanted. Established container grown plants, such as trees, roses, shrubs and perrenials can all be planted.