Thinking back to how I learned to sight-read....
Hmmm. I think what helped me the most was being forced to play songs I didn't know in front of an audience on a regular basis.
"Oops! The relief society pianist didn't show up, go drag Lybi out of young women's." Or "Lybi, will you go play prelude...it's getting too loud in here..." I think the most important thing about sight reading is to keep the music going without missing a beat. Playing with someone singing is a great way to hone this skill. You can drop non-essential notes here and there, simplify chords, drop the tenor line (sorry tenors) and no one will notice as long as you keep the melody mistake-free and the rhythm steady. Playing duets with a more experienced pianist is also a GREAT idea, Susannah. I used to do that with my sisters. Sightreading is actually a LOT more fun to practice than the early stages of a difficult repertoire piece, IMHO.
About improvising...experience with Jazz is very useful. If you took 10 years of ballet, you wouldn't expect to be perfect at hip hop or jazz dancing unless you had practiced it, right? But almost all dancers have experience in ballet because it is such a good foundation for all the other kinds of dance. Most pianist start with a classical background too.
A good jumping off point is to get used to a chord structure that is really easy like C, Am, F, G then back to C in the left hand (think "Heart and Souls") and then try to experiment with your right hand and see what pleases your ear. Eventually little motifs (little patterns) start popping up, and you can experiment with shifting it up and down the keyboard and turning it upside down and stuff. It's fun, too.