Do the milestones above apply to all games, regardless of what type of victory I'm aiming for?
Not necessarily.
Science victory obviously needs it, as well as Domination. If either of those victory types is what you're after, then follow Victoria's advice. Try to get to those thresholds if you can, but if not, don't worry about it since you're new. Playing on easier difficulties should help alleviate some of the pressure as well. Domination is the easiest victory type, and science probably 2nd. That's probably what you should be going for at first anyway.
For other victory types, science isn't as important. It's nice to have to keep up-to-date on your military, but playing on maps where you can be on an island helps make this less important. If you want a peaceful game, try Small Continents map.
Cultural and Religious victories focus more on the civic tree. A lot of what you
should be focusing on in the science tree early is unlocking Construction ASAP once you have all the basic stuff, just for the Games and Recreation Inspiration.
And the way I play Diplomatic victory you definitely don't need Science either (except for needing Construction, like I said, or the emergency Astronomy if you need Potala Palace) but even if you go for Diplomacy "normally" I can't really imagine a use for Science other than maybe keeping military units up-to-date . But if that's your only goal, it's not that hard. (Same can be said about Cultural and Religious victories.)
So knights by turn 80 is certainly not important for all victories. Furthermore, if you're going for Domination with your UU, then that should be your focus, not Knights (unless of course it
is Knights). And early wars might set you back, and barbarians can as well. You can't always have a perfect peaceful game.
And gold is not that important and very arbitrary. IMO, you should never focus on a set amount of gold. You want to focus on Food, Production, Culture, Science, etc. etc. and THEN gold. Use your gold to buy things quicker, and sell GPT in emergencies to get the gold you need now. Don't worry about what your current GPT is; it's not important. Many things can be converted to gold on a whim; luxuries, strategic resources, diplomatic favor, etc. Gold is just the "leftover crap" that you have extra. Think of it as a resource, not a threshold.
If you want more in-depth advice on Diplomatic victories let me know, as that's my area of expertise. Other victory types I can just give you vague information.