Well, I tried to make my answer as platform-agnostic as possible, but in that case, I'd work on the SEO to drive some serp traffic in, and maybe use those adwords coupons you have lying around to promote a landing page with prominently placed share links. You'll get a lot of mileage out of facebook links.
Probably depends on the kind of app it is. (I think we're assuming iPhone?) I'd say to start, make sure people can share it on fb and twitter everywhere possible. Then, share it yourself, spam your friends, write blog posts about it, contact bloggers and get people to write about it. If all else fails, spend some money, but hopefully it shouldn't come to that.
I've never done this, but my strategies would include: Create a landing page that ranks decently and drive serp hits to it, Make sure that the user knows how to use the system before they sign up (demo video), if you've got the money, run an adwords campaign for your target demographic, and try to make sure that your existing users invite others. Outside of that, friends, family, yayhooray, here, digg, put it EVERYWHERE. You can always limit the number of beta users if it's out too many places, and it'll help your ranking later if there's loads of backlinks.
Whenever I'm trying out a new language, I usually check if lynda.com has a tutorial on it. I happen to know that lynda.com does have one on PHP, and if you learn well with video tutorials, you can get to a pretty good spot with those tutorials in record time. If I remember right (It was a long time ago last time I looked at them), they don't get into OOP much, but if you're coming from C++ and Java, the php docs should probably be enough to help out.
Once you've got the basics down, the best way to learn would be to come up with a simple project and hack away at it. Build a tiny CMS or a blog, php is flexible, so you can come up with your own style pretty quickly, and then learn from your mistakes.