The thing worth knowing before anything else is that slim doesn't mean tight, and the moment you find a pair that grips you ankle to knee like a blood pressure cuff, walk away, because that's not the cut doing its job, that's the cut having no give left in it. A proper slim jean follows the leg without clamping it. There should be room to bend your knee on a train without the fabric straining, and if you sit down and feel your thighs being slowly rationed, you've bought the wrong size or the wrong jean.
If you've got none of these and need a starting point, go for a mid rise in a dark, plain wash with minimal stretch marketing on the label, something that just says cotton with a small percentage of elastane rather than anything boasting about "four way stretch", because the jeans that shout about their stretch the loudest are often the ones with the least structure, and they'll bag at the knee by lunchtime. A good pair has a bit of weight to the denim even when it's slim, a density you can feel when you scrunch it in your hand, and it springs back rather than staying creased. Cheap stretch denim feels thin and slightly plasticky, and you can usually tell just by rubbing the inside seam between your fingers, it'll feel dry rather than soft. Check the stitching on the pockets and fly too, uneven topstitching or a zip that catches going up is a decent sign that nothing else about the construction was given much care either. The button should feel properly riveted, not like it'll pop the first time you eat a big lunch in them, which, let's be honest, is the actual test of any jean.
Getting the fit right is mostly about where you try them on and how long you leave them on for. Sit down, walk about, do up a coat over them if that's relevant to your life, because jeans that look perfect standing still in a changing room mirror can turn out to dig in at the hip the second you sit at a desk. Watch the back rise especially, slim jeans have a habit of being cut high at the back and lower at the front, which looks fine from one angle and gives you a very unflattering gap from another. If the waistband gapes when you move but the thighs feel snug, you need a size up and a belt, not a size down and a prayer.
Can I sit in these for a whole afternoon without needing to undo the button under the table?