Hi Underbreath,
I feel your pain. I, too, prowl these forums looking for answers, and find myself wanting to ask: "So, just tell me- what do I need to get, and how do I do it?"
If you want to learn to seriously play the piano, you most definitely want an instrument with a weighted, hammer action. Casio, Yamaha, etc., offer digital pianos for under $1000, but they won't give you the synthesizer capabilities. You could, however, use such a piano as a controller for virtual or "soft" synthesizers in your computer. The musical creation possibilities here are almost endless, if you're willing to take on the learning curve involved. (I'm still clumsily trodding along this path.)
The difference between an arranger and a synthesizer makes for much lively discussion around here. It used to be that synthesizers were for creating vast arrays of musical sounds and effects, whereas arrangers were more for emulating standard voices like horns or guitars, but technology has changed all of that. If interface simplicity is important to you, then an arranger is probably the way to go. I found that I could passably imitate almost any musical style I wanted with a Yamaha PSR540, which you could no doubt find on ebay for a couple of hundred bucks (or less) nowadays. But, of course, you won't get the same sound quality you'd get on a higher-end, more expensive arranger. You also - for under $1000- wouldn't get the weighted keybed that is critical for actually practicing the piano.
Ultimately, what works for you is subjective, and for you to decide within your budget. There are quite a few folks here who know a lot more than I about specific workstations, synths, and arr keyboards, and can and will weigh in with their thoughts for you.
Welcome to the US, and good luck.
Cheers,
Michael
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Quasar