Whatever the founding fathers thought about states leaving the union, they didn't say much about it and the constitution says nothing. It doesn't really matter at this point, because the Civil War established a precedent that if a state leaves, it can only do so by the rest of the nation's agreement.
However, there is also precedent that individual parts or counties within a state can leave and form their own. West Virginia, for example, was formed out of Virginia from people within the state who opposed joining the Confederacy. Delaware was formed from counties out of Pennsylvania around the time of the Revolutionary War. Kentucky, Maine, and Vermont were all formed out of other states.
And there are many sections of states which have tried to form their own state over the years, right up to this day. Parts of southern Oregon and northern California wish to form a separate state called "Jefferson," so much so that Yreka (wye-reeka) California has a sign identifying its self as being the capitol of Jefferson.
Almost all of these would-be states are rural areas, tired of being taxed and controlled (and ignored) by urban areas. However, not all are. For example, the Florida keys declared they were seceding from the union to get rid of a checkpoint on the highway that was backing up traffic for miles and damaging the economy of the string of islands. Calling its self the Conch Republic, it even minted coins.
Parts of Arizona around Tuscon started a movement to become Baja Arizona because the state government was too right wing and refused to go along with an unhindered flood of illegal immigrants. South Florida has been complaining about those annoying non-leftists in the north for decades, and has proposed seceding from the state.
But the majority of state secessionists are driven by annoyance at being ignored and used by urban areas. Other sections of states which have tried to split away from existing states to form a new one include Absaroka (parts of Montana, Wyoming, and South Dakota), Superior (the Upper Peninsula of Michigan), McDonald Territory in Missouri, East Washington (basically everywhere but the Seattle-Tacoma area), Staten Island has threatened to secede from New York several times over being basically ignored by the NYC government.
On the other hand, Long Island has discussed separating from New York and forming its own state as well. Northern Maine is tired of their state being essentially run by Massachusetts and has discussed separating from the rest of Maine to form Acadia. Northern California discussed forming Shasta at one point, and Texas has had two major secessionist movements: to form Texlahoma out of parts of Panhandle Texas and western Oklahoma, and Jacinto out of eastern Texas in the 1800s.
Portions of Arizona, New Mexico, and Utah were discussing becoming the state of Navajo in the 1970s, and in the 90's the southwest section of Kansas tried to become West Kansas to get more funding and tax dollars spent in urban areas. During the build up to the Civil War, portions of Tennessee, Georgia, and Alabama did not want to secede from the US, and tried to form the state of Nickajack, but it wasn't as successful as West Virginia.
And then there's Lincoln, the state of western Washington, part of western Oregon and northern Idaho that has tried to split off for a while. A more rural southern New Jersey has tried to split away from the rest of the state a few times, and parts of Maryland have argued for splitting off from the state over taxes going to the cities and not helping their counties to form a new state, called various names such as Arcadia and Atlantis.
Some of these movements are pretty strong - the Jefferson one has been pushing for decades. These efforts usually come about from a long and angry disagreement with the state government, typically over taxes and anger over laws. Rural areas are taxed to pay for roads, subways, and other projects in big cities, which annoys them greatly. They tend to be more right-leaning and conservative in outlook, which puts them at odds with urban sections of the state politically and ideologically.
However, the chances of any of these movements coming to fruition are virtually nonexistent. States do have provisions in their constitutions allowing areas to split off, but they are difficult to carry out. The one furthest along at this point is a northern Colorado section made up of several counties such as Weld county and 10 others in rural areas. Mostly they are upset that a hard left legislature mostly controlled by the most populous city (Colorado) has rammed through law after law against the wishes of the rest of the state, such as homosexual "marriage," legalized pot, and various gun control laws.
Although the ballot measure may succeed in these areas, it isn't likely to go any further. All states require the state legislature to agree to a split, and since the splits are usually over annoyance at the state legislature, that's unlikely to take place. Further, states are loathe to give up a section of their territory because that means less tax revenue and usually these areas are vacation spots for skiing, hiking, camping, and so on.
And to make matters even harder, to form a new state, the US Congress has to agree. Since this would mean adding 2 new Senators, it becomes a political battle. Almost all of these potential new states would vote for Republican senators, and the Democrat-controlled Senate would just not care for that to take place. And if the state would add Democrat senators, well the Republicans would want to block it too.
And its expensive to add a new state. All the maps would have to change, and all the lists and information on the state involved and US data would have to be changed. The US flag would have to add another star - this time a bit more challenging, since 51 is a prime number and its not easy to squeeze one more star into a neat pattern. All the bureaucracy at the state and federal level would have to adjust to add another state, deleting counties from one area and adding them to another state.
All those county representatives, courts, jurisdictional efforts, law enforcement branches, and so on would have to change their location and coverage. Thousands of new highway and other signs would have to change. When Keizer split off from Salem, Oregon in 1982 it took years just to get their police force set up. The zip code for Keizer still lists it as Salem. And all of this adds up to millions of dollars in costs for the state alone, and many times that at the federal level.
Which means that political considerations aside, its almost impossible to get a new state formed. Still, its hard not to sympathize with residents of a state who are routinely ignored in the legislature, taxed for projects they see nothing of, get virtually no representation because of their sparser population, and continually are used by the rest of the state against their wishes.
And if one new state manages to pull it off, you can be sure there would be a half dozen or more attempts to get another one formed, with a great deal more momentum and likelihood of success.