Playing spin bowling is an art, an art which has undoubtedly been lost as time progressed. Our craze for hyping up the so-called "phast youngstas" has hurt our production of quality spinners.
Personally, I love batting against spin bowling, so from my own experience, it's mostly about watching the ball from the hand and just enjoying yourself manipulating the ball in different areas. Strike rotation is perhaps the most important aspect of facing spin bowling, because on most times, if you can pick up 4-6 runs from singles and doubles in the first 4 balls of a spinner's over, he will probably miss his length and allow you to drive or sweep him for a boundary.
Our batsmen play spin with stiff, locked wrists even when their intention is to rotate strike. That is the first mistake against any quality spinner, because if you don't have flexible wrists, you will be unable to guide the ball into gaps for runs.
Furthermore, on pitches when the ball is ripping and turning, the key is to understand when to be on the front foot and when to be on the back foot. Negotiating spin means that you need to be certain where the ball is pitching, because once you meet the ball before it lands, the whole element of the ball spinning is gone. Also, our batsmen need to learn how to guide the ball into good areas, on all sides of the wicket.
You can look at Babar and Rizwan and see how both are almost polar opposites against spin. Babar will try to push the ball into the V-region, whereas Rizwan will just make a deft touch and allow the ball to enter a gap. Both are good approaches for different situations, but you need to be able to do both in order to be successful against rotating strike against spin.
Now, hitting against spin is a different story. People say that using your feet is important, but our batsmen think of it in the incorrect way. Using your feet can be anything from charging towards the ball or even just moving on the back-foot to guide the ball away. In my opinion, unless the ball isn't turning at all, charging down the track usually ends better for the bowler, especially quality bowlers who watch the batsmen before delivering the ball. Once you've made your intention to charge down the track, the bowler who is smart enough to watch the batsman has a plethora of options to get you out. They could throw it wider and get a stumping, they could throw it between the bat-pad gap, they could bowl a literal half-tracker and get you to miss, or they could bowl a quicker, extra-full delivery and get you out. The risk-reward for charging down the track is quite poor, and though it looks beautiful when it works, it needs to be a calculated risk.
For example: the bowler bowls it fuller and you sweep him fine for four. He will now bowl it a bit flatter and more towards the off-side, whilst simultaneously pulling his length back, allowing you to advance and lift the ball over the in-field.
The slog-sweep is a great option against spinners in my opinion, it is minimal risk on wickets where the ball isn't spinning, and the only cause of concern is getting beaten in flight or by the pace on the ball.
One thing our batsmen have not learned is the reverse sweep. It is a great shot to get the bowler to bowl on your pads, because once he goes wide of off-stump, you can reverse him for an easy boundary and force him to bowl closer on the stumps, bringing the slog-sweep and conventional sweep into play.
The bottom line is that against good spinners, you need to be smarter than the bowler, and need to pressurize them into making mistakes. The best way to apply pressure is by picking out gaps for rotating strike, and then forcing the bowler to over-compensate to reduce the flow of runs, allowing you to find the boundary in a variety of patterns.
However, if you get stranded on the crease against a quality spinner, you can almost be certain that you will get out within the next few balls. That's the difference, that's the value of rotating strike. If you're waiting for the bad ball, you might get 7 runs from an over, but if you take singles and doubles on every ball, and subsequently punish the bad balls, you can get 10-12 runs and easily apply pressure on the bowler and opposition captain.
It's all a mental game.