Have you looked at Junonia for swim suits? They have a couple that I covet, and wish I could afford, because I want to go swimming sooooooo much.
You don't have to be ashamed of feeling nervous or scared about going into public. But, imo, you should think of a couple things when you do feel that fear: (1) If you're like me and many others here, you've been eyeballed/stared at and maybe even commented/jeered at before, and it hurts (or rolls off you, depending) and you felt the pain/anger and then moved on and continued with your life. You are able to do that again. You are able to handle it. And (2) so what if you're afraid/nervous? The presence of that feeling doesn't have to mean anything besides that you're afraid/nervous. Period. It doesn't mean you'll always be uncomfortable, it doesn't mean you'll never have fun, it doesn't mean a darn thing besides it's an unknown situation that makes you nervous for various reasons and that's scary.
My SO got me a couple books last summer, which I'm finally getting around to working through. One is called "Feel the Fear...And Do It Anyway." It's a cheap paperback, probably tons available used on Amazon, and I highly recommend it for helping deal with this kind of nervousness. It has helped me a great deal, with my own shizaam.
And as to those ladies bigger than you, whose "ease and comfort with themselves" you envy----have you talked to them? If not, you don't know what they felt going to the pool for the first time, wearing a swim suit and exercising, and you don't know what they feel right now when they go or when they're there. Do you know from their words that they are easy and comfortable with themselves? If so, she's a rare obese woman who is, then! lol Or do they appear easy and comfortable? Maybe they've been going for a while, and are, but weren't to begin with.
I think it's wonderful that you want to swim, and have the opportunity to learn and go for it. I think it's even more admirable that you're facing wanting to do it while feeling this discomfort, because that discomfort has prevented me from doing so many things (including swimming now, although the cost of the suits I'm willing to wear is prohibitive for me right now, also). If you're taking beginning swimming lessons, trust me, most people there will be way more concerned about themselves than you (not to be rude or anything, it's just that's the way it usually is)---how they look in *their* swimsuits, whether anyone will think it's "weird" for an X-year-old to be taking swimming lessons, moving in a new way in public, etc., etc. It's kind of like the first day of school---everyone's nervous! And you go and you learn and you deal with anything that happens, fun and not-so-fun alike.
Trust yourself. You can handle it, whatever "it" is.