All these recommendations are excellent. I'd say watch some shows from each decade from the 70s on. If you can find anything of the old Chicago Royal Airs from the 60s definitely watch and listen.
1970s
Santa Clara 1973 (their first title) or 74
Madison Scouts 1975 (as bold and powerful of a show you will ever hear)
Blue Devils 1976 and 1979 (real jazz on the football field and it was amazing)
27th Lancers 1979 (drill and color guard with A+ music arranging and you have a classic show)
Spirit of Atlanta 1979 (one of the loudest brass lines ever)
Bridgemen 1979 Civil War Suite (the South wins in Birmingham) Absolutely awesome show that would put people on their feet today
1980s
Blue Devils 1980 (professional grade soloists), 82, 86, 88 (more amazing soloists and a great reprise)
27th Lancers 1980 (many feel they should have won the title, no slight to BD who was excellent as well, but 27 had one of the best visual and guard presentations I had ever seen to that point with great music)
Garfield Cadets 1982 (an extension visually of what 27th Lancers and SCV had been doing, but to the next degree), 83 (Garfield's first title and the famous Z-Pulll), 84 (west side story, and perhaps the best interpretation of it that I have ever seen on the field. I consider this show one of the BEST SHOWS OF ALL TIME by any standard. It remains in my top 5 of ALL TIME shows), 85 (wicked hard for the time), 87 (ballet and grace)
SCV 1980 (one of the first asymmetrical drill moves, 1984, 85, 87, 88, 89
Suncoast Sound 1984, 85, 86, 88 (Florida Suite is one of my fav shows)
Madison Scouts 1983, 84, 88
Star of Indiana 1985 (Disney show), 86 (Space show), 87 (Circus show)
Spirit of Atlanta 1980, 84, 87 (high guard)
Bridgemen 1980, 81
Bluecoats 1987 (first Finals top 12), 1988, and 1989
Of course, they are all fun to watch. Many others to see too. Colts of Iowa were always great and still are. The Quad City Knights had some amazing shows back in the day, and I think 1984 or 85 really stands out.
The recommendations for the 90s and 2000s are all excellent, but I wanted to chime in on a few highlights from the 70s and 80s that had some historical significance.