There are some good websites out there with tutorials regarding the very basics of Street Fighter, but to start you off there are two ways to make combos happen.
1) Linking- This is when one basic move can follow the previous one and connect with the opponent before he had a chance to recover from the first hit. All of your moves in this scenario do their full animation and full animation recovery. The input windows can be very strict for more advanced links. An example would be jab, jab.
2) Canceling- This is what you're looking for in the above example. By inputting the motion and button press for a Shoryuken while the crouching HP is connecting, you will effectively cancel the recovery animation of the crouching HP (think when Ryu pulls his extended arm back) straight into a Shoryuken.
Also with buffering and other weird shortcut inputs, you should note that the Shoryuken motion actually has a shortcut of down-forward X 2, HP. Basically tap the lower right corner of the Dpad twice and hit punch.
EDIT: Sorry if that was too basic for you. You said you were a "complete beginner" so I assumed you knew nothing.