Lightroom is great for cataloging but it is also a very powerful RAW processor.
The latest version Lightroom 4 is very versatile and has very intuitive sliders and the processing does not take too long to learn.
The library side is rather more complicated but you should not be put off by this, get a decent book with it and you will soon learn it.
Remember RAW files are not an image yet, just data so you have a lot more flexibilty of what you can adjust with things like white balance, hue and saturation, shadows and highlights etc without degrading an image.
You can also crop, add graduated filters etc. Many people do not use any other software to process images. You also have a complete history of your developing so you can always undo what you have done.
Lightroom though is designed to go hand in hand with photoshop and you can export direct to this from LR.
Photoshop is better for manipulation than LR and is great if you wish to use layers (LR does not do layers) so you can make a new layer, edit, make a new layer, clone, etc and you can always hide layers or delete. If you wished to use advanced masking techniques to help work on a certain area without affecting another.
PS also comes with Adobe Camera RAW which actually uses the same processing as LR and does pretty much the same stuff actually, but that is the full blown Photoshop.
Plus PS does so much more than lightroom than I cannot begin to explain here.
Elements is like a condensed version of PS so is limited but still does most of what a photographer will need but you will be missing things like adjustment layers, but this is something not to worry about until much later if ever.
So, the choice? You need to decde what you arew going to do with your images, if you think you may wish to be creative do compositing of images or wish to go beyond basic editing then you will need Elements, if you think you will shoot in RAW and only do basic editing and colour adjustments / cropping etc then LR is more than adequate and you will not be disapointed.
Or get Lightroom first, learn to shoot and manage RAW files and catalogue and wen you are ready get Elements which is not too expensive.