Okey, i'll tell you a story how i would do it (generally method).
Assume that you have basic skills
So.
1. Open.
2. Copy layer with image need to be cutted (CTRL+J).
3. Copy one more time (CTRL+J).
4. Make Layer Adjustment Layer - Solid Color with some color. I usually choose red. Anyway choose color which summary differs from image (that will be background for checking cutting smoothness). Put it above background (so it will be something like that in Layer Panell - Layer 1 copy/Layer 1/
Color Fill 1/Background)
5. Now depending on what color need to be not there have to make choise, to make more contrast or not. Anyway its better do it usually. I prefer Curves. Playing with it (and if needed selecting some areas roughly) make color i don't wont different from others as posible. Don't care if image's colors goes total wrong (that for we have copied layer).
6. Depending of where unwanted color is choose selection method. If only background then usually its Magic Wand. If that colors is through all image then Select/Color Range. My default parametrs is Fuzziness:40, Selection active.
Press on some area with color want to select. If it is plain or preview you see is good for you then press OK.
Tip: You can select more by holding SHIFT and presing on some other area which is black in preview (will not be selected) on preview but you want it to be select (white on preview). If make more "selected" as needed can then press ALT+click to "deselect". Play till satisfied, press OK.
7. Selection apear. Now sometimes it is good to modified selection by Expand a little 1 or 2px (if image are small or have thin lines then don't). Feather sometimes is good to apply. A little again 0.5-2px depending on object which will be left sharpness (if it is sharp then don't).
8. Invert Selection (SHIFT+CTRL+I).
9. Now switch Layer 1 copy visibility off.
10. Select Layer 1.
11. Now or through menu Layer/Layer Mask/Reveal Selection. Or press that thing in Layer panell - box with white circle. That will create Layer Mask from selection.
12. So you see "cutted" image with some color background (from step 4).
And you can check is your cutting good or not.
13. If some details are not good then using brush with diferent size, hardness and opacity you can improve it by drawing on mask layer. Make sure mask is active (press one time on it). Drawing with black will make disapear "things" with white apear.
If everything is not good and painting mask looks that will take too much time then repeat selection process with some diference.
14. Now you satisfied with how much image is left and unwantable color is gone.
Delete all evearything except Layer 1 with mask.
You have your cutted-transparent image. Save it. Remember that TIF and PSD support transparency (in TIF you need to check that option).About other i don't know actually. JPG sure thing don't.
If you don't want to keep Layer Mask then just right click on it and choose Applay Layer Mask. But better keep it. You will be able to restore all picture just by Disable Layer Mask. With that method no information from image is lost.
In generally speeking thats all. Simple algoritm.
Some will say that it is too complicated but with those steps i'm sure that i will not damage my image and cutting will be good as possible.
P.S. If screenshots are needed then i can make them to real tutor (if someone is intrested).