Generally it is not worth using any card other than Gold to level, especially after a servant has already gained levels. This is due to the cost of using any ember is the same, while the exp given from a gold card vs. bronze is an extreme difference. It is best to use only matched cards as well for the bonus exp. I would also recommend that you save any ALL gold exp cards you get for any servant who has EXTRA class (Avenger, Ruler, and eventually Alter-Ego, Foreigner and Moon Cancer) though this is not for quite a while.
What's the best way to level up servants?
I know bronze embers aren't worth it and silvers are pretty much good for mana prisms. I just started playing a few days ago and should I just keep advancing the story until I consistently can farm gold exp cards and use those?
Play the event right now and get exp cards from the lottery is what I would say. 50 gold and silver exp cards per box, plus QP and some materials which should be enough to bump some servants up to 30 or 40 before ascension. The first couple of event nodes should be easy enough for even low level players, especially with a strong Saber Alter friend support. Also silver exp isn't bad for early levels servants, it's only later on when they are higher level and the QP and exp costs per level increase that it is better to burn silver exp for prism. There are other tips like only using half the exp required to get to your servants current max level since you can sometimes get a super or great success, which gives 2 or 3 times the exp you invested.
As a new player, I would say use all the ember cards you can get your hands on. But once you can consistently beat the 40ap ember gather quest, start sticking with gold.
The reason for only using gold exp is that it is the most qp efficient way to level servants. This matters because skill leveling is very qp expensive (5* skill lvl 9 to 10 is 20mil qp). But when you are new, all your ascension material is needed for ascension, not skills, so it's okay to be qu inefficient until you have a team to work with.
The story isn't really going to give you many exp cards. You get exp cards from the ember gathering daily quests, and from events. You actually should be able to handle the 40 AP ember gathering quest right now, since a decent support servant can basically solo it.
It's easy to use the daily quest to get a jump start for your servants, because the quest gives you so much master exp that you should pretty much always get a master level before you run out of AP, so it'll keep refilling your AP bar and you should be able to spam it. I wouldn't recommend that you spam it too hard, because you will want those free AP refills for some other stuff too. But it can be nice way to get a couple of servants up to level 40ish, so you can have an easier time starting out.
Otherwise, this current event is pretty nice for exp. I think we get like 450 gold exp cards out of it.