I am a swimming teacher and I teach adults as well as children and infants, I love teaching adults.
Congratulations on deciding to learn, swimming is a massive part of my life, I had a pool in my back yard as a child and have just bought a house with a pool. I spend every holiday at the beach, for me, swimming is pure joy and I hope that it becomes the same for you.
My first suggestion is that you head to the pool by yourself, head to the shallow end and get yourself floating. Initially, have yourself in the “starfish” position (legs and arms spread), ensure you have your stomach and bottom pushed up and have your head back (i.e. chin away from your chest). Relax, take nice deep breaths and notice how your body gently raises and lowers again in the water as your lungs fill and empty.
Teach yourself how to stand up from the floating position, bring your legs back together, bend your knees, push stomach & bottom down (i.e. bending at the waist) and bring your chin forward. Use your arms in a skulling action to help get yourself vertical again.
Next thing to teach yourself is breathing, I recommend you use a kick board (most pools have these available to borrow - at least where i am in Australia). Stretch your body out and start kicking. It is important to kick from the hips, as opposed to just bending you knees. The most effective kick has only a very slight knee bend, pointed toes, movement in the hips and ankles too. Think “floppy feet”
Put your face in the water, looking towards the floor of the pool, not infront of you and kick away. You need to make sure you are blowing bubbles in the water, and only lifting your head to inhale. Don’t hold your breath! Holding your breath with cause you to tire too quickly (you wouldn’t hold your breath going for a jog) and won’t help you relax either.
Have a go at my suggestions on your own before you pay for lessons.
You can probably tell this is my passion, if you want any more specific advise, send me a message and i will send you some lesson plans and or advice.
Regarding hair - I’m in the water 5 days a week, my hair colour doesn’t last, but i don’t care. If you have a shower and wet your hair first, it helps to keep it from fading - your hair is already soaked with tap water, so the chlorinated water doesn’t soak all the way in.
Good luck and have fun