Is a spider an insect or a bug?

No, Spiders are not bugs.They're arachnids.

Technically, spiders are not bugs since they are arachnids not insects. Technically only certain insects are bugs, belonging to the insect order Hemiptera. Therefore not all insects are bugs, either.

The answer is yes. Bugs are small arthropods by definition. Arthropods include insects, arachnids, and crustaceans. Spiders are arachnids, so they are arthropods. Therefore, spiders are bugs.