Public pools have a shallow end and a deep end. It’s difficult (but not impossible!) to drown in the shallow end because you can just stand up, but you can still swim.
Most humans, especially fat Western humans, are naturally buoyant. Completely inert, most (fat) Western people will float above the bottom of their nose (because we’re fat.) Very lean or muscular people tend to be more neutrally buoyant or even negatively buoyant (sink), YMMV.
Most important thing to remember as an Aquatic Mammal is you WILL get water in your nose, and sometimes down your windpipe. DO NOT PANIC. It burns, you will want to cough. Resist that urge. If you are under water or do not have free air passage, DO NOT COUGH. Control the urge and break the surface, then you can go ham coughing and sputtering.
The most important thing about being in and around water is to be comfortable. If you’re not comfortable, you’re too deep. Get shallow.