How to plant a garden on a hill and at the bottom of the hill?

Have you considered putting in raised beds? You could adapt to the hillside by shortening the legs/base.

I agree with jcraw. You would get a lot of water at the bottom of a hill so unless your soil is very porous you would most likely need raised beds or raised area to keep you plants from drowning.

Why not try terracing it? It would reduce the fall and you could put a path on each terrace to make tending your plants easy. Steps up to each level would make access easy. You could lay grass on the slope above each level.

I seem to envision a plan from all mentioned ideas... nearly flat at the house (be sure it drains away from house) and raised beds in steps going up the slope... the concept is like : https://www.hometalk.com/diy/outdoor/garden/timber-garden-stairs-steps-27052109

Instead of stairs.. soil for gardening. Even shrubs / trees! And I'd consider WIDE beds, not the (looks like 4' wide) narrow steps shown.

Further economy: the pics show 2x8 lumber... for raised beds you can use 1x8... HD has "ledger board" that is not fully finished (rough surface) and only a few bucks for a 1x10 x 10' . (Don't think ledger is a good permanent installation for stairs, that's different stresses. And a safety issue)