How can I get rid of this? Options

[HectortheRican]

How can I smooth this out? The 16-year-old glue from the emblems is killer.

[Galcobar]

General rule for paint -- start with the least aggressive method first.

A compound is, effectively, sandpaper in liquid/paste form. How harsh it is depends upon the specifics of the product and how it's applied. Improperly used, any abrasive product can damage your paint. Even properly used, they are designed to work by removing paint in order to create a level surface. If you can avoid thinning the clearcoat, do so (that said, a proper cleaning and polish of paint usually involves some abrasive action).

Personally, having removed a set of emblems from a 13-year-old Celica, I'd start with isopropyl alcohol. Rubbing alcohol is mostly isopropyl alcohol and will work, but often has other ingredients such as camphor. If you're buying go for the pure alcohol. If may do the job, but if it doesn't all you've done is strip any wax that might be on the area you've treated. In my case the alcohol, applied with a terry cloth towel, did get rid of most of the residue. I used Goo Gone for the isolated patches which remained, then applied a mildly abrasive polish (Meguiar's #80 Speed Glaze). Frankly, that area took less work to correct than most of the rest of the car.