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The first consideration are the proportions. 11 x 14 and 16 x 24 don't match the 3:2 proportions of your camera which will require cropping the image to fit the frame proportions. If you don't want to crop the images an alternative approach that I use is to enlarge the prints in 3:2 format then mat them to a larger standard frame size. That results in an asymmetrical mat but solves the problems. 3:2 crops in the range you are considering are: 18 x 12, 21 x 14, 36 x 24.
The next thing to consider is the printer's native resolution. That varies with brand and type of printer. Dry Creek Photo, which profiles Costco's printers, has a good tutorial on file prep here: http://www.drycreekphoto.com/icc/using_printer_profiles.htm
Let assume you decide make 12 x 18 prints at Costco. More than likely the printer will be a Noritsu 3xxx series which is 300ppi native resolution. So if you take 12 x 18 and multiply each dimension by 300 you get the needed image size in pixels (3600 x 5400) Dividing 5400 by your camera file dimension of 3888 the resizing factor is only 1.3, which minimal.
If you read the Dry Creek tutorial you'll discover the photo printers by default enlarge images by about 3% when printing to ensure there are no white borders due to paper misalignment. So it suggests "padding" the image to 3624 x 5436 pixels by first resizing to 3600 x 5400 then increasing canvas size to 3624 x 5436. You also need to tell the operator to turn off the automatic correction when printing.
I've followed the Dry Creek instructions and they have worked well. I resize the files to the desired print dimensions x 300 plus the canvas size padding suggested and put them on a CD which I hand to the operator with printed instructions. Dry Creek has a database of printer profiles by state which will alllow you to download the profile for the machine at the Costco nearest you.
If you use Photoshop you can download and install the printer profile on your computer and use it for "soft" proofing. Soft proofing is a simulation process which alters the monitor to look like the printer's gamut, less saturated in some colors and lower overall contrast. It's value is mostly in letting your know how the image will change in appearance when printed. During printing on any printer A 255,0,0 red in your file will be mapped to the deepest red the printer can produce and a 128,128,128 gray will also be mapped to a neutral gray tone per the printer's profile. So there's really nothing you can do to made the soft-proofed 255,0,0 red any more intense on the printer, but you can alter contrast and do other things like sharpening to compensate for the loss of contrast.
The Costco printers are not profile aware. What that means is the printer will manage the color per its calibration profile ignoring any gamut metadata (sRGB, AbobeRGB, ProPhotoRGB) in the file. How does a Costco print compare with an ink jet? The gamut of the printer relative to the editing space and other printers can be visualized with mapping utilities like ColorSync on the Mac.
Color Space is is plotted in 3D per Lab coordinates. Blue/Yellow and Green/Magenta horizontally and White/Black (Luminance) vertically. What you a seeing below is a side view of the Costco Glossy Print profile mapped inside of AbobeRGB editing space...
The parts where the colored Costco gamut hangs outside of AdobeRGB in the red colors represent that the printer's gamut can produce reds more saturated than the AbobeRGB gamut. So even with the Costco printer editing in ProPhotoRGB would be a better choice — the printer gamut should ideally fit inside of editing gamut.
The comparison below is between the Costco Glossy print profile and a Glossy print profile for an HP9100 ink-jet:
What is shows is that some colors will be more saturated on the HP (where the white wireframe is larger) but others such as yellows and reds seen hanging outside of the white HP9100 wireframe are more saturated on the Costco print.
All of this, in practical terms, means little. If you put either print on the wall individually, without side-by-side comparison, they would look good. It's only if you put one beside the other that you'd see the difference and since some colors are more intense in the ink jet gamut and others more intense on the photo print from Costco.
Given the fact that a 12 x 18 print at Costco is only a couple of bucks what I would suggest is prepping a couple of your files per the Dry Creek instructions, take them down to your local Costco, explain to the operator you've adjusted for printing without correction and have some test prints made. Best case you'll find they meet your expectations. Worst case you will have spent $5 and have a baseline for comparison with the results you get from other labs you use.
If you have a MacBeth color target a good way to evaluate changes from Capture > Print is to have your wife or kid hold it under their chin outside on a clear sunny day with the sun over your shoulder and the camera set to Daylight WB. Expose so the white patch on the target is 1/3 stop under clipping in the camera warning. Let it clip then close aperture 1/3 stop (one click). Convert the RAW file using ProPhoto RGB 16 bit editing gamut then resize to 12 x 10 and have a print made. Do not adjust based on monitor appearance.
On a clear day daylight WB in the camera should record the color neutrally. Keeping exposure 1/3 under clipping and face and chart out of any shadows should allow the camera to record a nearly full scene range. The file will be "nominal" out of camera. By not making any changes you eliminate the monitor and your adaptive color perception from the workflow and what you will see on the print when you compare it to the actual color chart will allow you to see, objectively, the best color you can reasonably expect from that printer.
If you print that same test file with familiar face and color chart on different printers then lay all the prints and the color chart together you'll be able to judge visually which you find produces the best results. It will cost you a few bucks to do that type of standardized testing but in the long run its money well spent because you'll understand color management better in practical terms and identify the printing method with the best mix of quality and cost that fits your needs. Depending on how critical you are the Costco prints might work for you, if not the most you'll spend is about $3 to find out it doesn't