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Is it true that in the high pressure compressor the stages and blades on the stages get smaller as it goes on?

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Yes, the volume of the annular air channel goes decreasing after each stage. When air is compressed it takes less volume. Each stage compresses a bit more, hence the volume is constantly decreasing. This is also true for the low pressure compressor.

In this cross-section, the two spools, low pressure in red, and high pressure in green, display clearly this progression.

enter image description here
Turbofan compressor section (GE90), source

The first stage of the high pressure compressor is longer than the last stage of the low pressure compressor, but note the annular space of the high pressure section is smaller than the low pressure section due to the mean diameter reduction. In addition the HP spool rotates at a higher RPM.

Engine from the same family, in 3D: enter image description here
Turbofan cutaway, source

The reduction of volume can be achieved by decreasing the outer radius of the air channel, by increasing its inner one, or by adjusting both. The number of blades/vanes can also be increased as pressure increases, to prevent deformation.

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