Hard to say, since I think more units in those armies fought through nearly every major campaign than not, especially Confederate units, as they usually served throughout the war while many Federal regiments mustered out mid-war after their term of service came to an end. But if I had to narrow it down to just a few....
The 1st Louisiana Brigade (Taylor's/Hays') played an instrumental role in Jackson's Valley Campaign and a few regiments that comprised the brigade were at First Manassas. They then served throughout every major campaign with the ANV til Appomattox (including the 1864 Valley Campaign), were always in the middle of the action and suffered heavy losses. But I think so many other brigades in Lee's army could also apply.
I haven't read about as many units in the AoP but, while not at First Manassas, the 20th Indiana was a prestigious regiment, though overlooked today. It was in the thick of just about every major campaign with the army from the Seven Days onward, with the exception of Antietam, IIRC.
In the Western Theater, Cockrell's Missouri Brigade was everywhere and was one of, if not THE best CS brigade in the theater. It was organized in early 1862 from Missouri State Guardsmen who volunteered for Confederate service - veterans of many of the early actions in Missouri like Wilson's Creek, Lexington, etc. They then went on to fight at Pea Ridge, were transported east of the Mississippi and fought at Iuka & Corinth, Vicksburg campaign & siege, then joined the AoT and fought in the Atlanta Campaign, Allatoona Pass, and were decimated at Franklin. The remnants went on to defend Fort Blakely, where overrun and captured. Also, the 1st Missouri Infantry was at Shiloh and later joined the Missouri Brigade in late 1862.
As for Federal regiments in the Western Theater, the 36th Illinois in the Army of the Cumberland had a pretty impressive record. It was at Pea Ridge, then at Perryville, Stones River and on through every major battle with the IV Corps, often in the middle of the action and always fought well.