I don't know enough about keyboards to make any suggestions, other than the weighted keys issue.
Regarding actual playing, there might be a few technique-related things I could advise. Something that's basic yet crucial is to keep track of your fingering. When you play a sequence spanning multiple consecutive notes, when sets of notes "jump" from one part of the piano to another, when single notes lead into chords, etc., you need to make sure you won't "run out of fingers" as you move your hand around. Admittedly, it's a little hard for me to explain, but you can see a basic example with scales. To play the eight notes in a scale you have to do something like this, from first note to eighth: thumb-index-middle-thumb-index-middle-ring-pinky. Considering how many ways you can put notes together, getting a feel for proper fingering is something that comes with time, but if you're learning one particular song you'll soon see what works and what doesn't.
There's really a bunch of things in piano playing that come with time, like the aforementioned fingering but also controlling volume and rhythm. Just keep at it and you'll see progress.