There's no easy way quickly master reading spin on a serve. In my personal experience, it takes many years to become adept at returning serves, even then its a weakness in many player's games.
Some tips:
1) practice returning serve against a higher level opponent. Have them serve their best serves. if you miss the serve, ask them to do the same serve again until you can get a good return on it. Start with one type of serve then move on to the next. For example (first underspin, then underspin+ left spin, then underspin+right spin, then side spin, then dead)
2) work on your footwork in returning serve. even if you can read the spin of a serve, if you're out of position on your serve return, you're not going to get a quality return.
3) focus on the contact and which way your opponents paddle is facing. easier said than done, especially when your good opponents can do the same stroke motion but contact the ball at different timing in their motion which imparts a different spin on the ball.
4) if your opponents are giving you side spin serves, aim to the middle. a bad return is not gettting the ball back on the table. a decent serve return is getting the ball back on the table, a great serve return is putting the ball where your opponent is not expecting
5) tactically, you need to pay attention to patterns your opponents are doing. take advantage of recognizing those patterns but dont get caught with your hand in the basket. you still need to watch the contact, get in position, and make a play on the ball.
6) if the serve is long, you can use your own power to overcome much of your opponents spin.
7) you have more time than you think on a serve return. typically, a serve is a much slower ball than a ball your opponent gives you in a rally, which means you will have more time to read and react.
8) understand the basic physics of spin....underspin serves with have a flat trajectory. topspin serves have an up then down trajectory. first you get in position to return, then you can react to the trajectory of the spin.
9) i just want to emphasize footwork one more time. i believe footwork is the most important aspect to being great serve returner.