GRD has already stated the main purpose of the tubes. I'd like to add some thoughts though:
The first two tubes (the 12AX7s) are used for amplitude (voltage) amplifiers. As soon as Cornet 2 has passive RIAA circuits the tubes also serve as part of frequency dependent dividers (R205-C202 and R213-C204 are low-pass filters, C203-R207 is a high-pass filter). The first 12AX7 together with the R200 also define the input impedance of the stage. The R203 and R207 define the bias points for the first two tubes. The R202, R209 - anode loads. C200, C201 are used for lowering power supply resistance for sound frequencies for first two stages and (together with R211, also dropping some voltage) are the part of additional power supply filter for sensitive input stages. R206 is a bleeding resistor - mainly for safety (to discharge C200, C201). R201, R208, R212, R214 - some kind of grids safety resistors.
The 12AU7 is also amplifier. In this case - not voltage, but current. As pointed by GRD - it is a cathode follower. The power supply for this stage consist of voltage dropping R220, R221 and power supply filter C206, C207 (shunt the sound frequencies path through the power supply too). Q200, R218 is a constant current source (CCS), defining the bias point of the 12AU7 and making the stage more linear. The R215 and C205 is just small current filter for the base of Q200 voltage which in turn defines the amount of the current which the CCS delivers ((6.2V-0.65V)/910Ohm=6mA, where 0.65V - is the base-emitter dropping voltage). The C208 and R222 form the output impedance of the stage (together with the output impedance of the last stage of course, which is very high because of the CCS).
