It's possible that you have water etching and you'd need to remove it by wet sanding and extreme buffing and possibly re-clear coating. However, with that said. Always start from the least aggressive to the most aggressive in paint correction.
The first step would be to do a FULL detail on it. Do you have a rotary? A DA? Have you ever used either of these tools? If not, than I suggest starting with a DA like the Meguiars G100 or G110 or PC 7242 and trying to polish the spots out with a paint corrective polish. As far as what product to use. There are a TON of products out on the market. I like Merzena myself but it is one of the higher end and more expensive polishes. #80 or #83 from Meguiars are both quality products and I use them quite often. Once you correct your paint which is going to take a few times, do a Iso. Alcohol wipe down to see if you really did remove everything. Once everything is removed than go onto glaze, seal, wax, etc.