You don't want to use Adobe's free reader for anything except reading PDF's. Acrobat is for reading/creating .pdf's, and is totally separate from Photoshop.
Even with Photoshop, you would save most files in .jpg format (equal quality, smaller filesize), or .gif at least.
Find out what software comes with the scanner, some are far better (Photosho/Painthop clones) than others.
MGI Photosuite is common, and not too bad a program for most everything you'd want to do (resize, crop, brighten, remove redeye, etc.).
Most packaged software will save (and recall) images based on date/filename, in folders using the same date/filename sequence, usually a subfolder of My Documents, but this can be changed within the program.
It's difficult to generalize, because there are so many image editors and twain compliant software around. Choose the scanner you want, find out what software comes with it.
Find 3 or 4 models, from different mfg's, and (providing they're all within a few $ of each other, and do the same job) get the one that is going to be easiest to use.
Post back here with your search results, and I can recommend the best software of the bunch (I've used a ton of different ones), and you can then decide if that scanner is the one you want.
Cheers,
Jim