When people post here seeking photo-manipulation software I normally direct them to Serif PhotoPlus 6. It's completely free and fairly easy to use:
http://www.freeserifsoftware.com/software/Phot oPlus/default.asp
However I'm not sure whether it will meet your requirements. (i.e. I don't know whether it permits you to cut objects from one image and paste them into another. It's still worth downloading though, because it will provide you with an easy-to-use program for simple manipulation of your images).
For your specific requirements, you might need a programme which uses 'layers'. These tend be less intuitive than programs like Photoplus 6. (i.e. you need to read some tutorials to work out what to do) . However they offer far greater flexibility. The best commercial software is Photoshop. The 'professional' versions, such as 'CS4', cost hundreds of pounds. (I'm lucky enough to have an older professional version of Photoshop on my PC. It's my favourite program). The 'amateur' versions (e.g. 'Elements 7') are cheaper but don't offer that much more than the better free software, such as the Serif program above:
http://www.adobe.com/products/photoshop/family /?promoid=BPDEK
Fortunately there's a free 'layer-based' program available, GIMP, which rivals the potential of Photoshop CS4. It takes a bit of getting used to but it's certainly worth downloading:
http://gimp-win.sourceforge.net/stable.html
(NB: Download the manual as well - all 21.5Mb of it! - and do a bit of reading before using the program).
Chris