This site has an excellent breakdown of everything you need to know about spin, including the various (27!) types of spin, the effects of the major types, how to read spin, how to handle it, and which types of spin are typically used for service or rallying.
I'm not going to list all of that here, but the author stresses that the angle of your racket and stroke direction are key:
Handling spin is mostly an exercise in racket angles and stroke direction. For every spin, there is a racket angle that will compensate for it. There is also a stroke direction that will compensate for it. Choosing which to use is the question. In general, use an upward stroke and open racket to compensate for backspin, while using mostly racket angle to compensate for other spins. (Open racket means aim racket upward; closed racket means aim racket downward.)
Against a sidespin, the more aggressive you are, the less the spin will “take” on your racket, and so the less it will affect you. The softer your contact, the more the ball will jump – so tentative players often have more trouble with spin than aggressive players.
When learning to read spin on a serve, it’s a good idea to wait on the ball, and take it as late as possible to give yourself more time to react. As you improve, you should start taking the ball quicker. However, even advanced players often take the ball later against a player with tricky serves.
The author lists ways to deal with more specific spins within their article.