Arrive early on a weekday morning if you can. The crowds thin out considerably, and the approach path has a quieter quality that's hard to find later in the day. The Kamo River runs nearby, and walking along it before or after your visit makes for a natural, unhurried way to get here or continue on toward Nishiki Market on foot.
Inside the grounds, take time to wander beyond the main hall. The smaller shrines scattered across the site include zodiac deity shrines that feel distinct from anything at a typical shrine visit.
At the main counter, buy your omamori, omikuji, and water fortune (水占い) all at once.