I am going to assume that the lawn/weed removal was not done very competently or thoroughly :-) Otherwise, you would not be facing the issue of at least the turf grasses growing back. And weed cloth has a well-founded reputation of being less than efficient in controlling weeds plus also seems not to have been handled properly to have weed penetration that rapidly.

You could use a herbicide if you were extremely careful with its application and avoided any desirable plants. Or you could go thorough and manually remove. I would also consider applying a good mulch once the weeding out was accomplished. Depending on the selection and type of the new plants, I'd consider river rock or wood chips. That would help to keep weeds down to a minimum but not eliminate them entirely - very few things will :-)) And any mulch will help to minimize soil moisture evaporation, reducing the need for any watering (although even very drought tolerant plants will need some to get established).

And next time, skip any weed cloth. It doesn't work well, makes it difficult to weed after the fact and retards the growth and development of other, more desirable plants.