Swimming lessons often start with the stroke, but what panics most beginners is coming up for air. Relax isn't an instruction. Breathing out underwater is.
Your first day
Starting takes standing in the shallow end with goggles on and repeating one thing five times.
Face in the water, blow out slowly through the nose the whole time, count silently to five, or until your breath is out, stand up and breathe in through the mouth. Repeat five times total. Then, only where you can stand up and with a lifeguard or friend watching, push off the wall and glide face down for a second or two, without kicking or pulling yet, still breathing out slowly.
Don't hold the breath underwater the way it feels natural to. Holding it is what causes the panicked gasp at the surface. Breathing out steadily the whole time underwater means the only thing left to do when you come up is inhale, calmly, because there's no held breath to burst out.
The thing most people miss
Much of the early panic in the water comes from breath timing, and less from the arms and legs, even though most teaching focuses on the stroke. Practise exhaling underwater and inhaling at the surface on its own, separate from any stroke, and swimming stops feeling like an emergency you're managing.
The strokes sit on top of a breathing pattern, and until it's in place they keep falling apart.
What not to do
Don't skip goggles to save money on day one, chlorine stings enough in the first week to wreck focus on its own. Water up the nose is the other real blocker nobody mentions, and it goes away by exhaling harder through the nose, not by pinching it or tipping the head back.
Once breathing is steady, the kick is the next thing to learn. The power should come from the hip, not the knee, and most beginners bend their knees too much.