Hey there, and welcome aboard.
A "banner" is the current selection of units that get an increased rate of pull from the summon gacha. As an example, the current "Developer's Choice" banner is for Gilgamesh, Abel, and Chizuru.
When the guide says to pull for banners and not for units, it means you should try and hold off from chasing one particular unit on a banner as it can be very costly to do so, especially if the one you are looking for is a 5* base (otherwise known as rainbows). They have a VERY low rate of getting pulled, even when their rates are increased due to being on the current banner.
So instead, wait until there is a banner that has more than one unit that is worth pulling for. There are even some banners in which EVERY unit available is worth pulling for, and that is ideally when you want to go "all-in" with your tickets and lapis.
As for your question regarding lapis... I'm not actually sure what the guide is getting at. I would say there are better uses for lapis than others, but please feel free to spend as you want to. "Smart" uses of lapis are generally to increase slots (as they are a relatively cheap, permanent benefit) and for daily pulls, especially when you are starting. Not so smart uses I guess would include some of the bundles (e.g. specific banner unit 2,500 enhancement bundles) and full price pulls (500 lapis). You will get upgrade materials and tickets aplenty by just playing, so holding off on those is recommended.
Oh, and I tried to add you as a friend, but it says you are full. (You are Griever, yes?) And if you want a GAF based friend list, make sure to add GAF to your player quote.