ています is the polite form of ている. On the other hand, てます is the casual and colloquial form of ています. The plain form 「ている」is also used in casual situation. How to know if it is better to use ている or てます?
ている is probably the most blunt form of the word. It'd be adequate if you are talking to juniors, or when you know the other person so well that the bluntness shows closeness.
ています is more polite, as you note. This is adequate, for example in the business setting, talking to somebody you don't know well,
ております is the humble form. This is better when you are talking to somebody whom you should show a formal respect, such as your boss, your teacher, etc.
てます or てる are casual forms. The former is a little more polite than the latter. These forms are adequate among your friends.
My advise is that if you are uncertain, always go for more politeness. It's a far smaller offense to talk too formal and polite to people who feel close to you, as opposed to talk too casual and impolite to people who expect otherwise.