Should I install gutters before addressing draining in the yard?
You can do the drain system and gutters/downspouts in either order. Assuming you are installing an in-ground drain system, you'll need to use the appropriate fittings to receive water from downspouts. You will want to install these when you're first burying the drain system, so you don't have to dig it up later to add them.

If you do gutters/downspouts first, you'll just make sure to route your drains so that you have intakes at the right spots. And you'll likely modify the ends of the downspouts slightly so they go straight down into the right spot.

If you do the drain system first, you'll have to place your downspouts such that they can flow into the drain intakes. Also pretty easy to do at installation time.

No matter what order you go, you should have a design for the whole system in mind before you start. The shape of your building and site may constrain both the gutter/downspout system and the drain system in different ways. For example, you may need a certain minimum number of downspouts to handle the expected rain flow.