My biggest suggestion is to soak the area in warm water (whether in the shower or bath) for at least three minutes BEFORE shaving. This relaxes the skin. It will reduce razor burn. The longer the better. Then, if possible, let the shaving gel or whatever else you're using sit for a few minutes. This penetrates the hair and makes it easier to cut.

Shave the finest hair first, letting the coarser hair soak longer.

Try to go over each area with only ONE stroke. If you must go over again, apply more gel/cream/whatever you're using. If you miss a hair, I suggest using a pop-up trimmer or tweezers instead of a second stroke of the razor -- you're less likely to get razor bumps and irritation.

Hold the razor at about 30 degrees to the skin.

Avoid shaving right after you get up. Your skin will be swollen and puffy at first. 20-30 minutes later, your skin will be more taut, exposing more hair.

Rinse your razor after EVERY stroke. Any hair trapped on the blade will reduce the razor's effectiveness.

If you have to apply pressure with the blade, that means it's too dull and should be replaced.

For soothing afterwards, a cool damp towel does amazing. Try not to itch. Also, try not to wear tight underwear, as it can rub at the hairs and cause them to ingrow. Wearing boxer shorts or something right after shaving instead of panties can make a HUGE difference.

Around your vulva, only use minimal ingredient products. Some lubricants actually do amazingly well as a moisturizer, amazingly enough. And I know they're safe around my vulva!