We just planted a bunch of beautiful mandevilla vines...we have these little bugs on them and they multiply. Does anyone know of a safe way to get rid of them for good?
Neem oil reapplied weekly. Someone will probably say hose them off with water but water doesn't kill them and they will probably find their way back to the plant.
Squish. Wash off. Or spray -- directs hits, please. (Insecticidal soap will work.)
And don't worry about them finding their way back to the plant. They aren't that mobile. You'll need to keep after them, I'm afraid to say. You never get rid of aphids 'for good'. Your goal will be simply to keep the population down to a minimum. Aphids have lots of predators and parasites to worry about, too, and if you do your part you shouldn't be bothered by them too much.
You might find that vine had too much N in the pot and will have fewer bugs once in the ground for a while (and provided you don't overfertilize). Most pests you never get rid of 'for good', but insecticidal soap made at home with a little veg oil and some cayenne (many recipes on this thread) will repel them for a while.