While this is a complex subject, today, it is easily implemented.
I'm not saying that my new Canon IPF 5100 is a better machine than yours is, because it's not, but, from the very first print, my prints matched my monitor WITHIN THE LIMITATIONS EXPRESSED BETWEEN EMISSIVE DEVICES AND REFLECTIVE DEVICES! Saturation will never be as high as what you see on the monitor, particularly in the greens.
Sorry for the caps, but this is often something people forget.
You may have fine prints, but are expecting certain colors to match where it is simply impossible.
For the most accurate work, you can soft proof as was mentioned. If that's set up and done correctly, you won't be so surprised at what you get. A good *free* tutorial on this is available from Adobe, on their site, conducted by the Digital Dog, one of our very own members, and a good guy to know.
But, even without that, there shouldn't be too much of a problem. While I'm not familiar with Hp's driver, or if they use a plug-in or not, as the Canon does, the concept is the same. The Hp is a very fine printer, with excellent matching capabilities.
As others have stated, use PS to manage the colors. The preference is for perceptual rendering for photo's, though you can go back and forth to check out whether relative colorimetric, which you are using now, is better with SOME images. It might be, it might not be.
You must set the proper profile in the printer (if the Hp allows that) as well as in the driver. There may be more than one place in the driver to put the profile, you need to do both, if so.
Always keep black point compensation on, no matter what anyone says. ANYONE!!! That includes paper manufacturers, printer manufacturers, etc. It does no harm, and may save your butt.
Make certain that the monitor profile is not set-up as your working space. It happens!!!
As long as you are SURE that your monitor cal is correct, this will make a big difference. Before using any self made profiles for the printer/paper combo, make sure you're using the correct ones from Hp, if you are using paper they supply. If you're not using their paper, I strongly suggest that you buy a package of Hp paper typical of what you use, let the printer do its thing with it, and work from that to make sure it's not the third party supplies that are the problem.
The monitor should always be profiled at 6500k, not 5000k, and gamma 2.2. 1.8 used to be the standard, but not now. The light source for prints, in theory, should be 5000k, but this is a big question.
First of all, having viewing lights that are the proper color, but are not in a booth with the correct wall color, are not accurate, no matter how good the lights themselves are. The prints should always be viewed while on an opaque surface, as light creeps behind a handheld print, changing the appearance. A thick while cardboard is fine if you must hold the print.
Secondly, make sure the lighting is really correct for the actual viewing conditions. In my own lab I had the entire print area, walls and ceiling, painted with the official Macbeth paints, which is a paint you can get from Pittsburgh Paint. The lights were Macbeth as well.
But for the customer areas, we used more convential lighting, because once they took the prints out of the lab, their lighting was uncontrolled. The print colors will seem to change, as you know.
I'll leave it here, though if you have any specific questions, don't hesitate to ask.