Thoughtful beats expensive here
A golf gift early in a relationship is doing a specific job: showing you noticed what she's into and took it seriously, without turning a nice gesture into a statement. That points away from big-ticket equipment and toward something well-chosen and used often. A dozen good balls or a smart accessory reads as attentive; a $300 rangefinder this early reads as trying too hard.
Pay attention to her game, not the price
The detail that makes this land is evidence you were paying attention. If she always complains about squinting on the back nine, a good pair of golf sunglasses shows it. If she plays a particular brand of ball, matching it exactly says you looked. A small personalized touch - her initials on a box of balls, a tumbler for the drive home - carries more weight than a bigger generic gift.
Steer clear of anything sized
Early on you probably can't verify her shoe size or glove fit, and a wrong-sized gift is both awkward and a little revealing about how much you actually know. That makes gloves, shoes and apparel the wrong call for now - stick to consumables and accessories that fit everyone. If you do know her glove size and hand, a leather glove is a lovely gift; if you don't, don't guess.
A pairing that always works
If you want it to feel like a proper gift rather than a single item, pair a small premium consumable with something personal: a dozen balls plus a marker stencil, or sunglasses plus a tumbler set. Two considered things wrapped together read as a gift with thought in it, which is exactly the note you want to hit.
What to avoid
Skip the joke gifts - the "World's Okayest Golfer" register is fine between old friends and slightly off between new partners. Skip off-brand balls, which any golfer spots. And skip clubs and putters entirely; they're fitted to her, and buying one blind is a gamble you don't need to take. For more picks in the gift-for-a-partner register, the golf gifts for her page has the full range.