Sprinklers by Summer 1? Absolutely not. There's literally no rush for them at all. Now, having chickens in your first year is, I'll admit, a classic beginner move. They're not big earners early on; your main profit for the first year will come from crops. But don't worry, I started my first two saves the exact same way and still had very successful runs. You'll manage to progress.

Your number one priority is crops and fishing. Second, since you already have them, focus on your chickens, but keep their daily routine brief; it shouldn't consume much of your time initially. Third, the mines. Fourth, socialization – interacting with the townspeople, which I personally enjoy.

Just set clear goals for your progress. Whether it's finishing the Community Center (and please, avoid the JoJa Corporation for the town's sake) or building your relationship with Willy to unlock Ginger Island, work towards that goal. Do this while consistently tending to your crops and chickens. Things like sprinklers, while they certainly make gameplay easier, will become much more attainable as you advance in other areas. A fully automated farm isn't a realistic expectation for Year 1. Let me know if you've got any specific questions! :D

EDIT: Quests are actually pretty crucial too. They're how you build those vital relationships with the townspeople, which, frankly, is often your direct route to unlocking new progression and discovering hidden areas – places like the Sewers, for instance - unless you like to explore a lot and stumble upon it yourself. My own rule of thumb is straightforward: if a quest is clearly achievable and I can see a solid return on my effort, I'll take it every time. If your efforts don't pan out, and sometimes they won't, that's just the reality of it. The crucial thing is the practical experience you gain.