A teenage woman who recently started following the laws of modest dress asked on Mi Yodeya how to balance staying modest with not embarrassing family members and others who are embarrassed by her "odd" dress. I wrote:
You are to be commended for taking on a socially-challenging mitzvah. It's not always easy to be Jewish and be seen as different, whether it's through dress, food, or how you spend your Friday nights and Saturdays.
With any observance that sets you apart from others, take care in how you talk about it. It's about you, not about them, especially for your secular friends. Make it clear that you're doing this for your own reasons; you need to steer clear of anything that will be perceived as a superiority attitude, because people don't like to be around people they think are judging them. Since that is not your attitude, this just means listening to how you talk and imagining how it will be perceived. One simple, truthful explanation you can offer is "I find these clothes more comfortable".
(Some of your friends are probably Jews who you might wish were also keeping tziniut, so perhaps there is some subconscious judgement there. Don't ignore them, but you need to get comfortable yourself before you can be effective at drawing others in. Being a role model will likely work better than trying to persuade them.)
Another thing you can do is to spend time with your female friends (without any guys around); in those settings the rules are more relaxed, so a swimming suit, for example, is not a problem. That won't work on beaches and at public pools, but it could work at a spa, for example.
Finally, if you act like this is (1) unremarkable and (2) not open for debate, eventually people will lose interest in arguing with you. If you don't make a big deal about it then, in time, they won't either.