Does the order of components going to ground count?
Figure 1. R3 and R21's ground points in what appears to be an audio pre-amplifier.


  ... but if I decide to let the signal go to ground in another point as shown in the schematic, would it affect the signal?



A schematic is not a physical wiring or layout diagram. It is supposed to show the schema or function and connections of the various componenents.
All of the GND points are connected together and usually by extra-wide PCB tracks so that there is very low resistance between the components connected to GND.
The PBC designer will use experience and design rules to minimise the likelihood of problems caused by poor layout. This generally means avoiding placing sensitive parts of the circuit close to high-powered parts where interference might result and placement of components such as decoupling capacitors C3 and C4 in their optimum position to do their intended job.



  If I take R3 and plug it to ground after R21 instead of after R2, it still goes to ground, but the order is different.


There is no order but you have probably increased the length of PCB trace into the circuit and this could cause problems such as interference pickup.


  Does it affect the signal? Does the output signal change?


If all goes well, no.