Like most of us, I played Lego as a child. With a handful of grand kids now, I've gotten back into it. It is still a great way to spend time together a be creative.

Up until recently, I've always classified Lego activity as one of two mode: Kit building following the instructions, or "free play", where you just build what ever strikes your fancy from what ever pieces you have on hand. I've come to discover an additional world of Lego where people make and share MOCs (My Own Creation). This might sound like the free play scenario, but what I wanted to share with you is the community of people and tools for designing, building, and sharing models.

Step 1: Resources

There are a number of places to find and share designs.

BrickLink, which is also a place to buy bricks
MOCHUB, where creators can sell custom kits.
Rebrickable, which also has alternate builds for Official Lego sets.

Additionally, there are people who design, sell, and share plans. Here are a couple of my favorites:

NK Custom Bricks sells kits and designs and has a number of free instructions.
JK Brickworks who's Sisyphus model was featured on Adam Savage's One Day Builds (a great watch if you want to see an adult share his unabashed love of Lego).

That is obviously just scratching the surface. Share your favorite in the comments and lets move on!

Step 2: Obtaining Parts

Lets say you have found a model you want to build. How do you know what parts you need and how do you get them? Every kit, including the official ones, comes with a part list, or a bill of materials (BOM). This gives you the quantity and color of the parts you need for the build.

Lego will sell you individual pieces using their Pick a Brick service.
BrickLink has resources for creating part lists and linking you to individual sellers of new and used Lego parts.
Ebay and Craigslist are good way to buy assorted pieces by the pound or lot. Search for "Lego parts".

Step 3: Part Nomenclature

In searching for parts, and even when building with friends, it helps to know how to describe what part you are looking for.

Element is the official term for a Lego piece. Its description will generally include the category (brick, plate, tile, etc.), color, size, and / or other information such as round, curve, or angle.
Brick is used as generic term for a Lego piece, but it more specifically refers to the traditional thick piece with studs on top. Additionally, you specify the size of the brick by number and pattern of studs in a brick. Probably the most iconic piece is the 2x3 brick.
Plate is an element that is 1/3 the height of a brick.
Tile is a plate that does not have studs on top.
Slope denotes a piece that has a flat surface that is not horizontal and is usually specified by the angle such a 18, 30, or 45 degrees.
From there there are of course many other special bricks. There is an unofficial term called "SNOT" or Studs Not On Top that is especially useful.

Step 4: Design Software

I've been amazed at the quality of the free software that can be used to build an document Lego creations.

Stud.io is BrickLink's free software. It has an instruction creator and can generate photo realistic renders. You can also upload the file to BrickLink to create wish lists to order the parts you need.
Lego Digital Designer is Lego's official software. It is no longer supported, but can still be downloaded and used in off-line mode.
LDraw is another free software than can be used to document builds and instructions.

If you want to try Stud.io, you can download my version of the F14 Tomcat at https://www.dropbox.com/s/sat8sz1wr8lii5z/Micro%20F14%20Tomcat%202.2.io?dl=0

You can also upload the file to BrickLink to create a Parts Wanted list. If you just want to see an example of the instructions you can created, I've attached a copy to this step.

Micro F14 Tomcat 2.2.pdf
Download

Step 5: Tips and Tricks

I am not an expert, but here are some of the things I have learned along the way. Feel free to share your ideas in comments!

Just because you do not have a specific element does not mean you are stuck. By their very nature, Lego elements can be combined to form what you need. For example, three 1x1 plates combine to replace a 1x1 brick.
Just because you found a piece in Stud.io, does not mean it is available. The more you stick to common elements, the easier it will be to source the pieces.
Just because you can build virtually does not mean it can be assembled easily in the real world. I will spend lots of iterations going between virtual and reality to make sure it can be assembled logically and won't fall apart when handled.
There are variations of elements that are more common that the original. The 3069b Tile 1x2, which has groove around the bottom is much more available than the 3069a Tile without the groove.