A painful bump is on my forearm and shoulder The one on my shoulder looks like it has a bite hole but the I can't see one on my other lump. Could these be spider bites?

They could indeed be spider bites, although there are lots of other biting insects and arachnids that could have caused them. As you'd imagine, insect bites are amazingly frequent all over the world. Most go unnoticed, and the vast majority don't cause any serious complications. Watch the bites you have. They should start to disappear over the next day or so. If they get a lot worse, call for help. There are some wonderful Poison Control sites (especially the one in California at UC Davis!) that can tell you all you need to know -- and more -- about bites.