Deserts are defined by their hostility to flora and fauna and a distinct lack of rain. Plains are simply relatively smooth areas of land. My question is how much area would be required to have a shift from a desert area to a plains area? I.E. If I started in a desert and walked in one direction, what is the realistic amount of distance I would have to walk to reach Grassy plains lands? If this is generally too broad then we can assume the dessert is a circle of roughly 4,000 square miles and anything past that border of about 33 miles from the center is no longer a dry and hot place. Could the grasslands start right at that border or would there be a dead zone where rain could fall and temperatures would be fine but little would grow?
Here an area of sand dunes is adjacent to scrubby grasslands. The dunes are held in place by a number of rivers running off the mountains. The rivers carry sand downstream, then when the rivers enter sink holes, the sand is deposited, and picked up by winds. This forms a region with a classic sandy desert appearance. The rainfall in the region does exceed the 250mm/year normally taken to define "desert" however, the high evaporation rate means that the sands are fully arid.
Beyond the rivers, shrubs and grasses can grow, even though rainfall is low, only about 280mm/year, this is sufficient for drought-tolerant grasses, and of course, the rivers bring additional water.
Flood Plains in Deserts
Consider the Nile flood plain:
The boundary between the desert and fertile land is very dramatic; to answer your question, grassland is literally a step away. Of course, the land is fertile because of flooding, not rainfall, so you have to decide whether this satisfies your criteria for grassy plains lands.
You could object that this is agricultural land, and not a pure grassland per se. Since the flood plain predates human agriculture, though, at least at some point in the past, it must have been a "wild" grassland along the lines you ask for.
Of course, if we accept that grassy plain does not necessarily require to be fertile because of rainfall, oases are the next obvious answer coming to mind:
Again, the fertile land is directly adjacent to the desert. Increase the amount of water available in the source aquifer, and you can increase the size of the oasis to suit your needs (the biggest oasis in the world is apparently the Al Hasa oasis in Saudi Arabia, 100 km²). Note that the height difference between the desert and the oasis does not need to be so dramatic as in the picture, although some difference is necessary due to the way oases form.
Alright I'll turn it into an answer. So first things first. What is a desert? It's basically a very dry barren terrain. This can range from sandy Saharra to rocky Atacama or Arizona. It can also go from searing hot to cold depending on location. The Gobi desert is known as a cold desert with the occasional snow even.
So how far do we need to go before we can find rain again or more important water? Not that far honestly. The aforementioned Gobi Desert is a rain shadow desert. This means mountains block rain for a bit. After that there is normal rain. This is in a way an extreme version of the dry side of a mountain. So on the mountainside you get a desert while in the valley below normal rain is falling. There won't be a clear border though. A desert isn't dry or wet. It's a scale, largely dictated by terrain height. I think the shortest distance between full grassland and full desert will either by a rain shadow from a mountain or another form of cliff catching rain all together. Either way it will be a higher terrain blocking clouds.
As I mentioned some deserts here are some grasslands. American prairies, African savannas, plaines in Central Asia and the Steppes in Russia.
Not far at all. Assuming your "grassy plains" defines the biome types of temperate steppe and grass savannah, there are places all over the world where those biomes abut either arid or semi-arid deserts or xeric shrubland.
In this picture above we are looking for yellow areas next to red ones. We can see this happens in multiple places in North America, Asia, and Australia.
In practice these will typically be transition zones, not abrupt borders. But you shouldn't have to go more than a few score miles to be able to tell the difference.