Getting a large single organism
The largest living terrestrial organism is the Aspen.
Each "tree" is actually genetically identical piece of the organism connected by root structure to each other. Aspen groves can be 5 miles or more on a side.
That said, I would say that a grass using some sort of root/vegatative propagation (aka Rhizomes) are the essential feature of plants covering a large area with a single organism.
Rhizomes provide the plant with the ability to cover vast areas with a single organism but provide no benefit to the plant developing intelligence or the ability to colonize.
Intelligence requires a lot of energy. The human brain composes just 2% of the human body mass but consumes 20% of the human bodys energy production. It is thought that intelligence's high demand for energy may require that organisms with high intelligence must consume other organisms high in protein and fat to supply enough energy.
Unfortunately, plants are even more energy poor than herbivores. Based upon the information in one of my prior answers (How much land area do ... herbivores need for food?), it takes at a minimum 1 acre of plant life to supply enough energy to keep 1 human fed. Presumably this 1 acre of energy production would be enough to keep the plants alive in the area but not enough for vigorous growth, competing with other plants, or intelligence.
Presumably your world girdling grass would require more land than this to harvest the energy it needed for both growth and intelligence. When the plant covered smaller areas, its intelligence would (based upon energy needs) be less. It would struggle competing with nearby plants that used all of their energy for growth.
As the plant grew in size, its intelligence could go up. I would guess that for human level of intelligence you'd need many acres of land (5+?) to develop consciousness.
Maintaining a single consciousness across distances
I don't know of any plant mechanisms for transmitting signals, other than chemical signalling. If this is the only method of signal transmission, then plant thought will be substantially slower than human thought - especially when considering the distance traveled.
I know of no method of transmission across uninhabitable portions of the planet (deserts, oceans, & mountains). You might end up with plant entities taking up entire continents. On Earth this might be 5 plant entities covering the continents (North America, South + Central America, Eurasia, Australia, and Africa).