It clicked for me at a wedding, standing at the edge of the dance floor with a tiny cross body bag that held my phone, a card and nothing else, while everyone around me was patting down their actual handbag trying to find theirs under lipstick and tissues. No digging, no missed photo because I was too busy rummaging. Just a strap across the body, phone in hand in two seconds, and I danced the rest of the night without once thinking about where I'd put it down.
The difference between a good one and a bad one shows up fast, usually within a week. Cheap versions have a strap that twists and won't sit flat, a zip that catches the lining, and a pocket so snug you're wrestling the phone out one handed while holding a drink in the other. A proper one has a bit of structure to it even if the material's soft, a strap that's stitched rather than just glued or clipped on, and some give in the opening so the phone slides in without you having to line up the corners like a jigsaw. Crossgrain or pebbled leather, real or faux, tends to hold its shape better than smooth, which scuffs if you so much as look at a table edge. Check the card slot too, if there is one, because a lot of them are cut so tight you'll bend your card getting it out in a queue.
If you've got none of these, start with a plain cross body in a neutral colour, long enough strap that it sits at hip height rather than under the arm, because anything higher starts digging in after an hour. Skip the ones with lots of extra pockets and buckles when you're starting out, they add weight and bulk without adding much use. A simple pouch shape with one zip and one card slot covers almost everything you'd actually reach for it during. These earn their place at festivals, on holiday where you don't want a full bag on a beach, at any event with dancing, and honestly for the school run too, when carrying a full handbag plus a child plus a scooter is not realistic. Summer is their obvious season but a good one works just as well under a coat in winter when you want your hands free for icy pavements.
The one mistake, buying one that only just fits your current phone. Cases add bulk, phones get bigger, and a bag that's millimetre perfect on day one will be refusing your phone entirely within the year.