You won't be happy with a rudder only plane. That was what RC flying was about 50 years ago. Those planes sucked back then too, but it's all they had.

$100 is a tight budget, but it can be done. Some of the small Parkzone "RTF" planes come complete for around that price. If you want to build it yourself, have a look at the Flitetest website for some info.

Cheap radios should have good enough range. They just won't have many programming features. If you can expand your budget a bit more, you can get a lot more for your money.

On the topic of budget, we'd be rude not to tell you that since this is a hobby, the cost is ongoing. You may hit your target budget getting your first plane in the air, but there will be setbacks and there will be the desire for more planes too. So don't think of it as a one time investment. Think of it as an ongoing expense for your enjoyment.