I avoided linen shirts for years because the first one I owned creased if I so much as thought about sitting down, and I decided the whole fabric was a waste of money. What changed my mind was trying on a proper one in a changing room and realising it hung differently from the collar down, less like cardboard, more like cloth that wanted to be worn.
Fit is where most people go wrong with these, because linen shirts are often cut generously to allow for the fact the fabric doesn't stretch at all, and that generosity can tip into shapeless if you're not careful. I look for a shoulder seam that actually sits on my shoulder rather than halfway down my arm, and sleeves that reach my wrist bone rather than stopping short, which happens more than it should once you're above average height. A boxy fit through the body is fine and often the point, but check the collar doesn't gape when buttoned, and that the shirt is long enough to stay tucked in if that's how you plan to wear it, since linen has a way of working its way out of a waistband by lunchtime. On colour, I've learned to trust the ones that forgive rather than flatter on day one. Pale oatmeal, washed white, a soft blue that's closer to chambray than sky, these all cope with creasing and a bit of wear far better than anything in a saturated colour, which tends to show every crease as a hard line rather than a soft one. White looks wonderful for about forty minutes and then shows everything, so if you want white, buy it knowing that's the deal.
The difference between a cheap linen shirt and a good one is mostly in the weave and the finish, not the price tag you'd assume. A thin, see through linen that creases into sharp folds and stays that way is usually cheaper linen woven loosely, while a better one has a slightly heavier hand, a density to it, so the creases soften into a texture rather than sitting as lines. Look at the buttons too, proper horn or a decent resin, not something that chips after a few washes, and check the side seams and hem for a actual finish rather than a raw overlocked edge that will fray the first time it goes through a hot wash. A good one also keeps its shape across the shoulders after washing instead of stretching out into something that looks borrowed from someone broader than you.
The real pleasure, though, is a linen shirt that's been washed enough times to go properly soft, the kind you can roll into a bag for a weekend and shake out rather than iron, and it still looks like you meant it.