Landscaping fabric or plastic should be put down below the mulch/rock.
Use a few layers of newspaper, or good quality landscape fabric and make sure to lay mulch or rocks at a depth of 3-4 inches to prevent weeds and grass from growing through.
You also need a good edge, or no matter what you do the grass will eventually encroach on your path.
If you allow leaves or grass clippings to settle on stone mulch, organic particles will accumulate around the stones and weed seeds will germinate there...despite the newspapers or landscape fabric. It will happen even sooner with red mulch.
I recommend applying a chemical weed preventer in March and September each year.
To follow up on Walter's note, look for a Pre-Emergent Herbicide like Preen or SnapShot available ate local home and garden centers. If you can find a local outlet of John Deere Landscape Supply, they will have the products that use professionals use.
if you want to have a stone type of walk way, first decide how you wish for the end result to look.
then dig a trench as wide as the stones you are using, and about 3 or 4 stones deep.
With any construction job of this kind you need a good solid base....without a solid base, your stones will eventually sink into the ground, will not sit flat, & will cause you all kinds of headaches.
There are many things you can use to give you a solid base, one that will (hopefully) outlast you. I suggest you speak with a contractor about what to use for a base.
Once the base has been laid & tampered/rolled, it is time to lay the stone.
Once you lay the stone to your liking, instead of worrying about the weeds, or landscape fabric, or any other traditional weed blocker, why not moss? It is a living organism that looks good, needs very little(if any at all)care, will keep ugly weeds from popping up between the stones, and is very easy to install/plant....
When you are out watering your plants, or grass, spray a wee bit of water on the moss. it won't hurt it....and if it gets walked on, that will not hurt it either.