The warning first: a coat only earns its spot if it fits over what you actually wear underneath, not what you tried it on with in a warm shop in October wearing a t-shirt. Sleeves that look fine over a blouse can go tight and short the second you add a jumper, and you won't find that out until the first proper cold snap when it's too late to take it back. That said, a good coat is the one thing that can make a rushed, nothing outfit look sorted, which is worth a bit of care at the buying stage.
If you've got nothing decent to start with, go for a wool coat in a plain, dark or neutral colour before anything trend led. It sits over everything, work clothes, weekend clothes, doesn't date, and a heavier wool holds its shape instead of creasing into something sad by the time you get off the bus. A trench is the second one to add, for the in between weather months when a wool coat is too much and nothing is too little. In the shop, do more than look at it on the hanger. Check the lining properly, run a hand along the inside seams, a coat with a cheap, slippery lining will cling to jumpers and ride up under a bag strap all day. Check the weight of the fabric between your fingers, too light and it won't keep wind off you, too heavy and it'll be a chore to carry around once you're inside and have to take it off. Try doing the coat up fully and then sitting down, a lot of coats that look sharp standing up strain across the back or gape at the button the second you're in a chair. Check the pockets are real and deep enough for your hands, not just stitched on for shape, and check how it sits when your arms are down by your sides, because that's ninety per cent of how you'll actually wear it.
A wool coat covers the proper cold months, school runs, early starts, standing at a bus stop in the dark, and it should still look right at a Christmas do over a nice dress if you get the cut right. A trench earns its stretch in spring and early autumn, those weeks where you need something over a jumper in the morning and off by lunchtime, and it folds over an arm without looking a state if you're carrying it. Neither one is built for a genuine downpour in a thin fabric, so don't expect miracles there.
Go up a size rather than down if you're between two, you need room to move your arms and get a jumper underneath without the seams pulling.