I think everyone hates me and thinks I'm not very good at my job. I think otherwise, but I'm worried this will effect my career. How do I shake it off?

Have a quiet word with your boss and ask for an evaluation. Ask if there is anything which you feel you could improve upon, or if you are doing things wrong. Potentially the way you were trained to carry out and complete tasks by another manager, (possibly at your current or previous place of work), is completely different from the way you're expected to do things here. Not everyone at work is helpful and will let you know this. Sometimes they will just sneer and look the other way, and pass you off as being ignorant. So you need to make sure that you are the better person here, and go out of your way to find out what the problem is an show that you are prepared to work on it and try and fix it before you are asked to leave. It could just be that some people at work are being really clicky. In which case, stay polite and try and get along with your colleagues - but blatantly ignore their sarcasm and bitchiness in your head :)