Photoshop has a dizzying array of hotkeys, shortcuts, and whatnot. Trying to remember every single one is not worth the time, as many of them are just a click away. However, for some of the most used commands in Photoshop, there are CTRL + ALT + SHIFT key combinations, regular menu access combinations, and more.

FILE MENU:
New document: CTRL + N
Open an existing document: CTRL + O
Save: CTRL + S (Warning! Saves over the existing file; if the file is new, then the Save As.. dialog will open. To prevent overwrites, use the Save As... shortcut.)
Save As: SHIFT + CTRL + S
Close current image: CTRL + W (If file was edited, it will prompt to save)
Close all images: CTRL + ALT + W (Again, if any files were edited, you will be asked if you want to save the changes)
Print with Preview: CTRL + P
Exit the program: CTRL + Q

EDIT MENU:
Undo the previous action: CTRL + Z (Depending on your settings, successive presses will either undo the undone action, or undo the next previous action)
Cut: CTRL + X
Copy: CTRL + C
Paste: CTRL + V
Free Transform your image: CTRL + T
Stroke: ALT, E, S (Stroking is PS's word for outlines)
Fill: ALT, E, F (Fills the selection with the color chosen in the dialog)

IMAGE MENU:
Image Size: ALT, I, I
Adjust Levels: ALT, I, A, L, or CTRL + L
Adjust Curves: CTRL + M
Adjust Color Balance: CTRL + B
Adjust Hue & Saturation: CTRL + U
Invert Colors: CTRL + I (Not to be confused with Invert Selection!!)

LAYERS MENU:
New Layer Via Copy: CTRL +J (A selection must be done before this can be used)
Merge Layers: CTRL + E

SELECT MENU:
Select All: CTRL + A (Selects the entire image)
Deselect: CTRL + D

TOOL PALETTE:
All of the tools can be accessed by using the single key-strokes listed in the tooltip for that button. To change to another tool in the smae button, type the letter for that button, followed by SHIFT + THE LETTER to acces the other tools marked with that letter (the possible tool versions are visible by hold-clicking on the triangle in the bottom-right of the button). After changing the tool version, there's no need to repeat the SHIFT combination to use the version again, unless you want to use the original version. For Reference:

Marquee: M
Elliptical Marquee: SHIFT + M
--------------------------------------------------
Move: V
--------------------------------------------------
Lasso: L
Polygonal Lasso: SHIFT + L
Magnetic Lasso: SHIFT + L
--------------------------------------------------
Magic Wand: W
--------------------------------------------------
Crop: C
--------------------------------------------------
Slice: K
Slice Select: SHIFT + K
--------------------------------------------------
Healing Brush: J
Patch: SHIFT + J
--------------------------------------------------
Brush: B
Pencil: SHIFT + B
--------------------------------------------------
Clone Stamp: S
Pattern Stamp: SHIFT + S
--------------------------------------------------
History Brush: Y
Art History Brush: SHIFT + Y
--------------------------------------------------
Eraser: E
Background Eraser: SHIFT + E
Magic Eraser: SHIFT + E
--------------------------------------------------
Gradient: G
Paint Bucket: SHIFT + G
--------------------------------------------------
Blur: R
Sharpen: SHIFT + R
Smudge: SHIFT + R
--------------------------------------------------
Dodge: O
Burn: SHIFT + O
Sponge: SHIFT + O
--------------------------------------------------
Path Selection: A
Direct Selection: SHIFT + A
--------------------------------------------------
Horizontal Type: T
Vertical Type: SHIFT + T
Horizontal Type Mask: SHIFT + T
Vertical Type Mask: SHIFT + T
--------------------------------------------------
Pen: P
Freeform Pen: SHIFT + P
--------------------------------------------------
Rectangle: U
Rounded Rectangle: SHIFT + U
Ellipse: SHIFT + U
Polygon: SHIFT + U
Line: SHIFT + U
Custom Shape: SHIFT + U
--------------------------------------------------
Notes: N
Audio Annotation: SHIFT + N
--------------------------------------------------
Eyedropper: I
Color Sampler: SHIFT + I
Measure: SHIFT + I
--------------------------------------------------
Hand: H
--------------------------------------------------
Zoom: Z
--------------------------------------------------
Default colors (B&W): D
--------------------------------------------------
Switch background/foreground colors: X
--------------------------------------------------
Switch between Standard & Quick Mask Mode: Q
--------------------------------------------------
Standard Screen Mode: F
Full Screen with Menu: F
Full Screen without Menu: F

These are the most commonly used keystroke combinations in Photoshop. There are five tools that CANNOT be accessed via the SHIFT + LETTER combinations:
Single Row Marquee, Single Column Marquee; Add Anchor Point, Delete Anchor Point, Convert Point.

Another great shortcut is holding CTRL and clicking the layer name in the Layers window to Load Selection.