Do I need special primer for red(ish) paint

Not a problem. I have used any cheap water based paint as a primer. Primer has two functions. Seal the surface so you use less of the high price spread, and give tooth to the surface for the finish coat to stick. You don't have to keep it all waterbase or all oil base, unless the paint can directions are specific.

The paint/primer should be fine, especially with two coats, but I typically begin with primer, as it seems (in my experience) that primer holds paint better and lasts a bit longer before requiring touch ups or repainting, and it's often cheaper to do a coat of good primer (match latex to latex, or make sure you follow the label if you use a different combo), then two or three even coats of less-expensive paint (paint+primer is almost always 2-3x more expensive than paint alone.

I wouldn't sand and re-paint, of course. You should be fine with what you have so far.

[Edit: one other note on paint+primer paints; make sure you read reviews or ask others about their experiences with the particular brand you're buying. A friend ended up with a patchy-looking paint job when he used one of the cheaper brands. I don't know if it was an isolated incident or if the brand is normally bad (or if he didn't mix the paint well!), but research before you buy.]