Power rail decoupling caps for analog circuitry
All active components should have decoupling capacitors. The PCB traces between your part and your power source act like parasitic resistors and inductors, and if you don't decouple your ICs, then when their power requirements change quickly - for instance, when they try to change their output signal in response to something - the changing power requirements will cause voltage drop and overshoot due to the long PCB traces. A nearby decoupling capacitor eliminates that high frequency noise by providing for those short-term spikes locally.

0.1uF capacitors are a reasonable default for decoupling; if your device may have particularly large power draw requirements, you should add a 1uF or larger capacitor in parallel.

Regarding grounding, you will find a lot of conflicting advice on this on the Internet. Split ground planes are often less simple and more problematic than you might suppose, because return currents prefer to flow on the reference plane underneath the signal trace. If any of your traces cross the split in the ground plane, you will force that current to deviate around the split in the ground plane, causing a lot more noise than you might have eliminated by splitting the ground plane in the first place.

This article provides an excellent description of why you should consider using a single ground plane combined with careful routing.