Is it good to have a higher slew rate? What are some of the effects due to slew rate limitations?
Slew rate limits will cause distortion at high output frequency and amplitude.  If your amp is slew rate limited, putting in a periodic waveform (sine, square, etc.) will result in something that looks a bit like a sawtooth wave.  This can create frequency harmonics that are not present in the original signal, especially when the source signal is a pure sinewave.  Generally, you need to have a high enough slew rate for the highest frequency and output voltage that your circuit needs to support.  

The slew rate is another term for slope.  The maximum slope of a sinewave is equal to its amplitude times its angular frequency (derivative of \$A \sin(2 \pi f t)\$ at the zero crossing at \$t = 0\$ is \$2 \pi f A\$).  So a 1 MHz signal at 1 V amplitude (2 V peak to peak) has a maximum slope of \$2\pi \times 1 \text{V} \times 1 \text{MHz} = 6.28 \text{V} / \mu \text{s}\$.  If your amp has a slew rate of less than \$6.28 \text{V} / \mu \text{s}\$, then you will get a triangle wave if you try to get it to output a 1 MHz 1 V sinewave.  Note that the slew rate has to do with the output voltage of the op amp, not the gain.  That being said, it usually affects high gain circuits more because the signals tend to be larger.  

In op-amps, slew rate and bandwidth tend to be linked - high speed op amps tend to have fast slew rates, otherwise they wouldn't be very useful.  Fast slew rates will allow an op amp to overshoot or ring at a larger output swing than an op amp with equivalent bandwidth but a slower slew rate.  Slower slew rates can help limit overshoot and ringing in many cases.  Another thing to consider is the power supply - the output current has to come from somewhere.  Very fast slew rate op amps require a very low impedance power supply.  This may require placing multiple capacitors of different values very close to the op amp - generally a combination of large, bulk capacitance and small, high frequency bypass caps .  

Slew rate limitations can be helpful for reducing the harmonic content of digital signals.  Some devices have a propensity for producing very fast edges (e.g. FPGAs) which, while necessary for high bandwidth communications, can cause problems with lower speed communications.  Fast edges can couple to adjacent traces and can cause crosstalk and intersymbol interference.  Limiting the slew rate can mitigate this.  Transmitting serial data over a limited bandwidth (e.g. for an RF link) also takes advantage of slew rate limiting to confine the bandwidth of the signal.