Photoshop or Lightroom
The case for Photoshop
PS is a very rich bitmap and photo editor - it can get at the RAW data if that is what you want by depending on an intermediate module like ACR or LR. It has very many tools, and the skillset you pick up with that interface brings you to a professional level that can be used to make money or seek employment as a photo manipulator or a graphic artist.
PS came from photography but also from the art world. You do not have to begin the art piece with a photo. You can make an art piece using PS from zero.
It deals with photos at a dot by dot level. It has a huge list of tools and subsequently effects that allow you to adjust the whole picture, specific regions or single dots (I think). It allows you to layer and mask the picture like a real painter with a real canvas and real paint could do.
PS Elements I think lacks macro recording so you can't record a script and play back the effect on another photo. PS CS5 can but the price is quite a bit more. LR can record a "look" based on the parameters and batch apply it to a whole bunch of other photos.
If you don't want a fat product or an onerous payment per PC (legally), use a competitor like Corel Paintshop Pro - which has a different culture and way of working, in some sense, tutored and simple - but with less engineering refinement. It is not expensive and the Ultimate package comes with some goodies.
The case for Lightroom
LR started off as a RAW reader PLUS. The plus has now been enriched with more and more features inclusive of cataloguing. It is parameter based not so much pixel by pixel based. It is probably a king in being able to draw non proprietary information out of RAW and along the way, tweak the look BEFORE handing over to PS.
Lightroom came from wanting to firstly read the RAW file and then apply overall, parameteric changes to the picture. For example you want to apply some brightness or some sharpening, very much like how Picasa works, so LR is very good at that.
Simply LR lacks the complication and thus even I could make a photo look better by adjusting a slider or two. With Photoshop I don't even know where the sliders are and which slider is better to use or am I in completely the wrong screen.
If you don't want a fat product or an onerous payment per PC (legally), use a competitor like Corel Aftershot Pro.
PS vs LR
If you want to be able to read and understand the thousands of PS articles, how-toes, magazines, web info, videos, learn PS.
If you want to be able to, with the minimum of fuss and not a deep and experienced understanding of techniques, make a picture better, learn LR.
The case for neither
Both are Adobe products - fat in size, fat in price and with Adobe activation policies and licensing.
If you don't want to pay (in one sense, be able to walk to any machine and carry your editor with you without worrying about technical issues and licensing) use Portable GIMP, PAINT.NET (installed but I have seen a portable version) and RAW Therapee.
Remember, your camera also comes with software for photos editing.