Costas Papaconstantinou already pointed this out, but I highly recommend that you create your logo in Illustrator, not Photoshop. Unless you are 110% sure that your design will never need to be scaled up (which is highly unlikely), Illustrator is the way to go. That said, if you do decide to create your logo in Photoshop here are the tools I recommend you focus on becoming comfortable with.
I suggest working in Shape mode (A) when using your pen tool. Unless you're comfortable visualizing the positive and negative space within your paths and their color, it's far easier to work in Shape mode. Your paths remain editable and you can work with fills and strokes in a similar manner as Illustrator. But perhaps the single most important tool you'll need to get comfortable with is the Pen tool itself (B). The best way to learn this tool is practice, practice, practice. You should also get used to using the keyboard shortcuts associated with the different tools in the Pen Tool palette and all the other shortcuts that will make editing and manipulating your paths quicker.
While using the Pen Tool (P):
OPTION/ALT – Convert Point Tool: This allows you to make a point into a corner, break your handles so that they affect your path independently of one another, and turn corners and/or independent handles back into smooth handles locked to the same length and direction.
COMMAND(Apple)/CONTROL(Windows) – This will enable you to change your Pen Tool into the Direct Selection Tool (white arrow) without having to switch back and forth between tools. Moving points, changing handles, and changing curves while still remaining in the Pen Tool will enable you to work much quicker.
SHIFT – After closing your path, if you want to start a new path/shape from any of your points or along your path just hold down Shift. If you're not trying to start a new path/shape from a point or along a curve, you don't need Shift; after the path is closed you can just start your next path/shape without doing anything.
(Subtracting and adding points to your path/shape don't require any keyboard shortcuts. Just hovering over a point with your Pen Tool selected will change the Pen tool to the Delete Anchor Point Tool, and likewise just hovering over a path with the Pen Tool selected will change your Pen Tool to the Add Anchor Point Tool.)
Not to be confused with changing your Pen Tool's paths to Shape mode, the Shape Tool (A) gives you some preset shapes to choose from. This will save you the time of having to manually create things like Rectangles, Rounded Rectangles, Ellipses, Polygons, Lines, and Custom Shapes, and you can also use these as your base shape and build your logo or elements out of them. The Custom Shape Tool (B) allows you to create shapes based on various preset images. There are different libraries to chose from (C) and if you want you can create your own custom shapes and custom libraries.
(This isn't really central to logo design, but just in case you want to know how to create those custom shapes)
Select your shape layer in the layer panel and change your tool to the Direct Selection Tool (white arrow). Control click (right click) on your image and choose 'Define Custom Shape' (A). Rename your custom shape whatever you want (B) and it'll be added to the open library.
Depending on how you decide to work and how well you can digitally paint in Photoshop, the Brush Tool (A) can be an important tool to understand. I highly recommend purchasing a pen tablet if you find a need to use the Brush Tool often (I use Wacom). It's possible to use this tool with a mouse or trackpad but it is significantly easier with a pen tablet. Familiarizing yourself with all of the Brush's panel options and settings (B) can take some time but there are a lot of tutorials out there that cover pretty much everything. In my opinion the only reasons to use the Brush Tool in logo design are for sketching and fine or hand drawn details, but then, I'm not that great of a digital painter. I'm not sure I'd ever find myself using the Brush Tool when finalizing a logo but you never know when you might need it to paint in small, complex, or hand drawn details. As far as I'm concerned the most important function of the Brush Tool in logo design is for sketching logo ideas. A lot of people work with pen/pencil and paper for their initial design ideas, but with a pen tablet, Photoshop's Brush Tool, and a little practice, you can achieve the same thing (in my opinion, faster). My advice would be to use Photoshop only for your sketches and then move on to Illustrator for your final design.
CLIPPING MASKS / LAYER MASKS
Having taught myself Photoshop, I spent years without ever using Clipping Masks. One day I just stumbled into a tutorial that relied heavily on them and my whole Photoshop world changed. Any layer can be used as a Clipping Mask by either Option/Alt clicking the line between the layer you're looking to make a Clipping Mask and the layer underneath it, or by selecting the layer you want to make a Clipping Mask and selecting 'Create Clipping Mask' from the layer panel's drop down (A) in the upper right corner of the panel or by right clicking the layer. In the above example I was looking to paint a soft blue edge on the lower logo element (B). (This is the way it would look before making the layer into a Clipping Mask, but you should make the layer a Clipping Mask before painting)
As you can see that same blue brush area, once it's layer has been made into a Clipping Mask, only appears in the alpha of the layer below it (B). Once made a Clipping Mask the layer will display an arrow pointing down and shift a little to the right in the layer panel (A). There are a lot of uses for Clipping Masks and once you've become comfortable with the concept you'll find it an invaluable tool for more or less everything in Photoshop. The same can be said of Layer Masks.
Any selection can be used to mask out portions of your design with Layer Masks. You can make selections with your Lasso Tools, Marquee Tools, from paths, and by using the alpha of any layer (Command (Apple)/Control (Windows) Click the layer's image icon). Once you have a selection you can click the icon at the bottom of the layer palette (A) to create a Layer Mask on the active layer. This will result in a mask icon next to your layer's image icon in the layer's palette (B). You can unlock and lock the mask (click on the little chain link next to it's icon) so that it moves independently of the layer (unlocked) or with it (locked). When your mask is selected you'll have various options to control it in the Properties panel (C). The important thing to remember about how masks are represented in that little mask icon (B) is that white represents fully transparent and black represents fully opaque. This is important to remember because with the mask icon selected on your layer you can use your brush to hide or reveal parts of your image by painting in either white, black, or anywhere in between.
If you're using Photoshop for your logo design it's important that you work large (i'd recommend at least 300dpi and large enough to cover all possible applications). But the key to designing a successful logo is to create something that works at all sizes. A good way to keep a reference for how your logo will work at smaller sizes, while working large, is by using the Navigator panel (A). The Navigator gives you the ability to see your entire comp in a small window, and if you choose, you can use the Navigator panel to zoom in and pan around your comp. Having a small reference of your logo while working is a good way to ensure that your creation is not becoming illegible at small sizes. Depending on the complexity of your logo you'll sometimes have to make a few different versions so that it remains visually consistent across all sizes. Some people duplicate their comps in order to get a view of it small, but the Navigator seems easier to me.
While I highly recommend using Illustrator if you're working on a logotype, Photoshop's Type Tool (A) can be used as well and everything in the Character panel (B) is similar to Illustrator's. If you need to manipulate a character's design you can convert your type to shapes by right clicking on your type layer in the layer panel and choosing 'Convert to Shape' (this is similar to converting your type to outlines in Illustrator). Once your type is converted to shape you can manipulate it's paths like any other shape with your Pen Tool.
Explaining everything there is to know about color in Photoshop would take a very long time so I'll just explain what everything is. The color Mode you work in (A) depends on what you plan on doing with your logo. In all likelihood you're going to need a bunch of different versions for a number of formats (web, print, broadcast, etc.). Keeping your colors consistent across formats is a question that maybe someone else could answer better than me, so I suggest making it a separate question.
The basic Mode selections are, RGB for web and broadcast, and CMYK for print (B). Depending on the format you're working in you may want to set your Color panel (C) settings up for that format. When working for the web you may want to set your Color panel settings (D) to 'Web Color Sliders', 'RGB Spectrum', and 'Make Ramp Web Safe' (that last one isn't necessary as your color selections will be converted into Web Colors anyway). If you're working for broadcast I'd recommend you change your settings (D) to 'RGB Sliders' and 'RGB Spectrum' (broadcast safe colors could be an entirely different topic so I won't go into that other than to say that you can load an action in the Actions panel (Video Actions – Broadcast Safe) that will convert your colors to broadcast safe colors). For print you'll probably want your setup to be 'CMYK Sliders' and 'CMYK Spectrum' (there is a whole lot to know about print colors but you should ask someone else since I work mainly in broadcast). There is another option though when choosing color in Photoshop and when designing logos it's an important one.
Regardless of how you choose your color in Photoshop, I suggest that your final color palette should be made into swatches and saved. Swatches are essentially preset colors. You can either select your colors first and make them into swatches or you can create your color palette out of swatches (to make one of the colors you've chosen in your Color panel a swatch just make sure the color is in the front (D), click in an empty space in the Swatch panel (C) and then name it what you want). The top line on the Swatch panel (A) is essentially a swatch history bar; it keeps track of the last 16 swatches you've used and constantly changes to update your history. Beneath that are all of your swatches. By opening up the Swatch panel settings (B) you can choose from a whole bunch of different swatch collections including various Pantone and Web collections (the Swatch panel settings (B) are also how you save your swatches).
(I forgot to mention the basic color setup on the left-hand side of the screen. D and E are your front (D) and back (E) colors and they can be switched back and forth by pressing X, or by using the little arrows next above them. Your front color is the color you're currently working with. F is the black and white preset for your colors. Clicking that will return your front and back colors to black and white, respectively. You can do the same thing by hitting D)
There are of course a lot more things you can use in Photoshop when creating a logo, but these are more or less your basics. As I mentioned above and as anyone else will tell you, you really should be doing all of this in Illustrator. As far as learning how to design logos goes there are a ton of tutorials on the web that will teach you the fundamentals of logo design. Learning the design principles and operating the software are entirely different subjects, but now you have a brief explanation of some of the tools you can use to execute your design with.
Hope this helps.