Lightroom, or its alternative, is pretty much like a wet "darkroom" in the old days of film. You edit each pictures, tweaks levels, adjust colors, contrast, crop, dodge/burn, etc. Some software are more sophisticated than others, but essentially, that's what it is, and you can proceed in batch.
Photoshop on the other hand can do pretty much all you would do in Lightroom, but would let you work at a much higher level. You can repair pictures, add or remove objects, use layers, etc. There are also a lot of plugins available to enhance pictures. Photoshop is a great piece of software, but there are other alternatives that may suit your needs just as well.
If your intention is only to take pictures with your camera, do some basic editing so your pictures look nice and you can have some of them printed, then all you need is Lightroom or one of its alternative. Actually, you can download ACDSee Pro 8 and try it out for 30 days free. Best way to make yourself an informed opinion is to try it out.
I'm old school, and I don't like the new licensing model that software companies are putting forward where you "rent" software for a monthly fee. I prefer to pay up front, get to use the software for as long as I want, then upgrade whenever I feel like doing it. I don't know if it's true, but I've read all your files made on software rented monthly can no longer be opened if you stop paying and decide to no longer use the software. If so, you may end up screwed pretty badly in the future if your computer is full of files, but you have stopped using those "rented software".