@WisteriaHysteria Grenade spam
I remember at first every class had a very large number of special grenades (nail grenades, MIRVs, etc) but then a patch reduced it to "1" I think. One team would just spam those grenades on the doors to the enemy's respawn point, you'd get wrecked, and then in future matches you would be adopting that strategy yourself and encouraging your team mates "go go go, grenade their spawn". It felt like magic when you managed to lob a deadly grenade *into* the spawn point, and not just at the doors.
The "Hunted" maps had one player controlling this chubby executive/president guy with an umbrella (a crowbar weapon reskin) that the rest of the team of soldiers, medics, and heavies had to escort against a team of snipers. I remember we had all these memes about the president, what hitting someone with your umbrella meant, and so on, but it has escaped me now.
Mastering the art of the "solo" play was definitely a thing. It depended on how good the enemy team was of course, but if you stuck to a class you'd learn very good strategies for each map. It'd always be a countdown until one of the good players on the other team wises up and starts tackling your shenanigans head on. There were websites like planetfortress which would have "guide" pages and linked forum posts on how to play each class. I remember reading these strategy pages the same way I read about strategies for Starcraft.
The team effort going on was always this "push, push, push, rah, rah, rah" feeling in the forefront of your mind. If you could see an enemy player wrecking house you'd make it a mission to prioritize them to help your team, because taking matters into your own hands was always more reliable than just typing out a team message and hoping for the best. Sometimes it felt like you were the first person to pick up on an enemy player's plan (by observing them trying to do something with each death), so you felt especially responsible for inserting yourself into the situation to stop them.