First I will like to point out the differences in swimwear for guys.
Boardshorts are swimsuit that are loose on the skin. They do not have any nesh brief sewn in on them. They also typical don't have an elastic waistband to them. They also are longer in length then swim trunks. Typically they can be worn either to swim, surf, or hang out at the beach with no problems and they can be worn just as shorts on a hot day.
Swim trunks on the other hand have a mesh brief in them. They also have a elastic waistband to them. The length of them are usually shorter. Most of the time they are worn for swimming and on a beach or pool deck but are not usually seen as an alternative to shorts.
Swim briefs also known as speedos are a brief style swimsuit that is used mostly for competitive swimming and diving. In other countries they are used as a primary swimsuit. Some guys like to wear them in more public places.
Swim Jammers also known as jammers are a cross between boardshorts and swim briefs. they are worn for competitive swimming. they are more modest in there length and are seen more as a public option.
square cuts are like breifs and jammers as they cover more than the brief but still have the shorter length than the jammer. They aren't as common as swim briefs and jammers but are highly stylish.
Rashguards are shirts that are used in swimming and water activities. They are more recent invention of some sort. Made of a material that doesn't let uv light through. they have become a weapon of choice for parents who want to protect their children from skin cancer.
They are many other things that are used as swimwear but these are the basics.
Now here are my tips for choosing they right swimsuit.
First concider what you are going to be doing most. Like if you are going to be on the beach mostly then a boardshort or swim trunk will be better. but if you are going to be in competive swimming then either a swim brief, a jammer, or a square cut will be better and you can decide from there based on other swimmers.
Next concider your body type. If you are on the chubby side a swim trunk might not be a good choice as they have an elastic waistband which will cut in. but a boardshort also usually doesn't have a mesh brief in them to keep the goods from popping out. Now if you are on the skinny side you can go for almost whatever you want. As far as height the longer legs of boardshorts go a bit better when you have long skinny legs.
Next see what you like better. go to a store that has different types of swimwear and place to try them on. but my advice is to get as many different types as you can and see what works for you more. once you have found one type you like you can get more of them.
another tip about swimwear underwear is not to be worn under them. underwear of any type is not designed to handle water very well. chlorine in pools will damage underwear and cause them to fail early. underwear also doesn't dry out as quickly as swimwear so they will be wet a lot longer causing shrinkage and chaffing.
one last tip is to wear sunscreen when outside no matter your skin tone. but you also should get some natural sun rays to boost your vitamin D.
for parents your son should be allowed to go shirtless in the summer whenever he wants to and for your daughters let them wear a bikini it is not a sexual thing to be in a bikini. and really if you allow boys to go shirtless then they girls should have an equivalent to that.