I think you will find it is simpler than you think.
Firstly, according to the PS8810 user manual, page 41, 1 Amp of current is available from all outputs. However, page 107 does mention 10mA and 1A in the next sentence, meaning that with a 10mA load, 10V will be present. I think you can ignore that 10mA. 10mA would only be enough to drive the smallest of reed relays or just the base of a transistor, both of which are not much good at switching real world loads.
Secondly, with relays you need to look at what load you are switching to determine the size and type of the relay. Relays do not provide an output voltage, they only provide switching contacts for that voltage. You must provide a power source to be switched, and you can switch any voltage as long as it is below the rated contact voltage (usually 240VAC for a power relay, which is probably what you want) and rated current. Therefore in your case you may need a 24V power supply to be switched by the relay, for your equipment.
As for the coil voltage, 10 volts is usually enough to operate a 12V relay. I've found really cheap brands may buzz and not switch on properly at 10V but a good 12V relay should activate at 8-9 volts if supplied with ample current (generally if its a good relay it must fully activate at 80% of it?s rated coil voltage).
Anyway, an Omron LY2N-12VDC works perfectly with a USM810. If all is the same on an 8810 it should be no problem also. Omron usually deals through RS and Farnell and just about any industrial automation company will stock Omron. It is a 2 pole (it can switch two circuits, eg active and neutral) but is available in 3 or 4 pole also. The datasheet is available at http://ecb.omron.com.sg/pdf/relay/general/LY.pdf
These relays have contacts rated at 10A. You may get away with much smaller relays depending on your current, however I find these ones extremely reliable. Their major advantage is that they can be mounted on DIN rail with the Omron PTF08A plug in sockets, which also provide *beep* terminals ie no soldering. If you have ever tried to neatly mount a PCB relay in an audio rack, its not easy. They also have a built in LED which lets you see if the relay is energized or not.
As for solid state relays, they are really only needed for nasty reactive loads (for example switching for power amplifiers) to avoid the arcing which occurs across relay contacts when there is an inrush surge due to the power supplies in modern switching amplifiers. Arcing will cause the contacts to weld then the thing cannot be turned off. Solid state relays by definition don?t have contacts to weld. But if you aren?t switching heavy loads, (which you probably aren't if you are switching 24V) then you shouldn?t need to spend $$$ on solid state relays.
Hope that helps.