Shrinking image without losing quality?

Photoshop can indeed change the resolution (to a certain extent) without ruining an image. The trick is to uncheck the "resample" check box in the resize image tool. Resampling the image is what creates the issues you have found.

But if all you are doing is trying to print an image, you don't need to do all that. Print the image at the defined size you wish, and any decent printer from the last 10 years will do so with no further input from you.

PPI and DPI and all that mess are pretty much irrelevant. You tell the printer you want the image at 4.5x6" as an example, and the printer does the rest. Provided you have the printer driver set up properly for the paper type in use, there is nothing else you need to do.

Pring the original by using "fit to page" or defining the print size in the print settings.

That will give you the maximum possible quality for the image you have.

Paper print quality should ideally be 1200 pixels per inch for really high quality work.
Screen display only needs a tenth of that to look OK.

You cannot shrink an image (in pixel dimensions) without losing quality.

Sounds like you reduced the size too much. What you want to do is first determine that size of print that you want to make, and then multiply the resolution by 300. For example, if you want a photo-quality 4"x6" print, your file must be at least 1,200 x 1,800 pixels. Anything less than 1,200 x 1,800 will not be photo quality and will look pixelated.

you cant shrink a pixel -

the quality of the print is always defined by your printer - the higher its resolution (pixels/inch) the better the print you will get when you shrink it

all your software resamplers do it "throw away" pixels from the the original photo

(Say you want to reduce a photo by 1/2

what they do is look at a group of pixels (say 8x8) then calculate what a group of 4x4 should be (colour wise) to look the same as the 8x8

(try shrinking one then enlarge the smaller result to the original size - then compare this with the original)

What you are describing is perfectly normal.

The general rule is to print an image to its natural print size. If you have a decent graphics program, you should be able to convert the size units to inches or cm to see exactly the size your photo will print to. BTW, the minimum DPI of printers is 300 so if your photo is less than that, convert it to DPI 300 first before you check the print size. If you are seeing only pixels, then make sure DPI is 300 then simply divide width and height separately to DPI to get inches. Following that, naturally, if you shrink an image, you won't be able to print to the original size anymore. Beware, making the image larger also has the same effect but for another reason.

Don't shrink the image - just increase the dpi - thereby maintaining the size in inches and the quality.