The National Guard didn't exist as such prior to the end of the Spanish American war. Before that, it was called the Organized Militia, although some units called themselves National Guard. Up through at least the beginning of the Civil War militia units generally required members to purchase and own their equipment including firearms and uniforms. There was also no standard uniform and units adopted their own based generally on what the Army was using.
In addition to the Organized militia, there were units of Volunteers formed in time of war. These were scratch units formed usually by wealthy individuals and outfitted in various ways and as various types of unit but mostly infantry or cavalry. Some more famous volunteer regiments over US history include:
The Kentucky Mounted Rifles in the War of 1812. The governor of Kentucky called for men with means to provide themselves a horse, rifle (not musket), etc., and volunteer for service. The regiment reached about 1,000 men at its height and proved to be one of the best units the US had in the Great Lakes region.
Wilder's "Lighting" Brigade. John Wilder, an Illinois lawyer, using regiments of volunteers devised a means of pursuing and defeating Confederate raiders / cavalry using men riding mules as dragoons (mounted infantry). He paid for his brigade to be equipped with Spencer repeating rifles--a huge increase in firepower--as well. A Confederate curse at the time about this weapon and its fearsome firepower was That damned Yankee rifle! You load it on Sunday, and it shoots all week!
Roosevelt's Rough Riders would be another from the Spanish American War.
These are Volunteer militia rather than organized (or standing) militia like the National Guard.