Photoshop is a huge, powerful piece of software. I've been using it for 15+ years and I still learn new things about it every so often. I have never used Photoshop Elements, which I understand to be a pared-down version of the full monster. So bear in mind that these comments are based on the full Photoshop.
If you want to teach yourself to use Photoshop, there are tons of how-to books out there. Just jumping in and playing around with the filters and the healing brush is fun, too. But if you really want to get comfortable in P'shop, here are some things I advise you focus on mastering. Not just playing around with.
To be completely honest, upwards of 75% of the time I spend in P'shop is just selecting the pixels I want to do something with. It can be surprisingly difficult. There are dozens of ways to make selections, but these are the ones I've found most useful.
- Pen tool - Take the time to get good at adjusting those Bezier corners. They're flexible and very precise -- the pen tool draws a vector path, not a line of pixels. This skill will serve you well in Illustrator, too.
- Quick mask - Not to be confused with masks in general. What quick mask does, basically, is allow you to use any of the brushes, gradients, etc., as selection tools. Using gradients for selections is very useful for making gradual changes to an image.
- Lassos and the magic wand - They have their uses and they can be quick, but in low-contrast or complicated images they can become frustrating. Of them, I use the magnetic lasso most -- but it needs a high-contrast edge to work best.
My only regret in being a print-oriented person is that I was trained to CMYK color correction. I can look at a photo and tell you which of the four primary colors is off and how to fix it.
The online world runs on RGB, though, which is a whole different universe that I do not understand. So I'm advising you to start with RGB and build up experience with color correcting in that gamut.
The main tools you want to master for any color correction are Curves and Levels, which are under Image>Adjustments. You can apply these individually or as adjustment layers so that you can tweak them later (I recommend the latter. See the next point for why.)
Layers and masking
The Layers palette is your friend. Imagine each layer as being a sheet of clear plastic that you've laid over your original picture. You can paint on it, make adjustments, mask, etc., and then you can turn it on or off at will.
One thing you'll learn quickly in P'shop: you want to keep your options open. You want maximum flexibility, and layers give you flexibility. (Putting your color corrections on layers is included in this.)
Masks are pieces of art that block out parts of the picture below -- like actual masks do. P'shop masks can be large or small, sharp-edged or fuzzy, transparent or opaque, and they can blend in a variety of ways. Masks are one reason that you spent so much time mastering your selection tools, because once you have that selection you can turn it into a mask and play around with that bit of your picture with impunity.
Masks are applied to layers. They are independent of the layer's content, though, and can be edited separately (or together) for maximum effect. It's very much worth your time to learn to think of masks and layers as separate things that interact with each other as well as with the underlying image.
And now you can see how easy it is to go on and on about P'shop. I spewed all this out in about half an hour, and I didn't even go into any real detail.
If anyone's interested, I could do the same for InDesign, but that's not a program that anyone's going to buy on a lark and play around with.