How is this managed when there are several shafts connecting several compressors/turbines?
The different shaft will rotate at different speed from each other, but the relative compressor/turbine pairs will rotate at the same speed.
Expanding a little, in the most complex case that I heard of, there are 3 shafts: High Pressure (HP), Intermediate-Pressure (IP) and Low Pressure (LP)
HP turbine and HP compressor will rotate at the same speed, but this is not necessarily equal (in fact is highly likely that it will be different, otherwise there would not be the need for different shafts) to the speed of the IP compressor/turbine couple. And this will be different also from the speed of the LP turbine/fan couple.
Are reduction boxes needed here or do they, in fact, move at the same speed?
They generally rotate at the same speed. And generally you don't want gear boxes in the central section of your engine. Apart from the maintenance nightmare that they would become, given the rotational speeds and energies involved, they would be extremely heavy, inefficient and would constitute a critical system quite prone to failure.
An exception are the fans, they have enough space and are in front of all the rest of the engine, allowing the housing of a gearbox within their enclosure. You can find a better discussion about this case in this question; keep in mind that here you have 3 compressor stages with only 2 turbine stages, the gearbox was necessary to have a speed differential between fan and IP compressor without requiring 3 turbine stages and an additional shaft. (thanks to @fooot for pointing the PW1000G out)
Is this a big deal, does it introduce any serious limitation/drawbacks regarding the exiting air power?
As everything in engineering, there are trade-offs. The ideal performance would be obtained via an infinite set of infinitesimally thin turbines/compressors, each rotating at a slightly different speed, but this is not technologically feasible. The engine designers will then settle for a slightly sub-optimal performance, but it will be actually achievable with 1 or 2 (3 in extreme cases) discrete shafts.