1. First, choose your doodle or sketch. (If you draw it in pencil, you should go over it with a fine liner ink pen - preferably black so it will be clearer when you scan and live trace it.) Scan it into any graphics program and Place it in Illustrator by going to File > Place.
2. Now we want to Live Trace it, so select your drawing and go to Object > Live Trace > Tracing Options. In the dialogue box choose Simple Trace from the Presets drop down menu and check the Ignore White box. Then click Trace and Expand it by selecting Expand from the top toolbar.
3. As you can see, some of your lines will be quite messy so we'll have to tidy them up. Use the Direct Selection Tool (or white arrow tool). My general rule for keeping the lines clean is to have as few points as possible. So try and spend some time editing your points. Here's my before and after:
4. Now create a New Layer (should be Layer 2) and drag it underneath Layer 1. Select your image and copy it into Layer 2. Then lock Layer 1 and hide it (click on the eye icon). See image:
5. Now select your image from Layer 2 and go to Object > Live Paint > Make. Select the Live Paint Bucket from your Tools panel on the left and choose a color. Click in the unfilled areas to fill with color. Ignore the outline for now. When you're happy with what you've got (again, you can add more detail later) select your image and Expand it again. Then Ungroup it (you might have to do this more than once). Now, select your outline - this can be done with the Magic Wand tool from the Tools panel and delete it. You should end up with something similar to this:
6. Unlock Layer 1 and make it visible (click on the eye icon again). Select your entire outline and change the color. Now, if you zoom in a bit, you can see some white bits showing through.
7. We need to get rid of this by making the outline slightly larger. To do this, select the outline and make it a compound shape (Object > Compound Path > Make). Then go to Object > Path > Offset Path and enter these setting values (Offset: 0.2 pt, Joins: Round, Miter limit: 4). This makes a copy of the outline underneath the original. Now you can delete the smaller original.
8. And there you are, from sketch to a nice clean vector.