This answer is heavily based on this article. I'll summarise it here, but please read the full thing.
It might be very tempting to start designing in Illustrator right away. Don't. The first thing you want to do is sit down and start making decisions. What type of font (serif or sans), what is the targeted use (body text for long documents, headlines, decorative, ...), what will be unique about it, ... There are a billion things to decide before you start.
Before using the computer, you might want to draw some characters by hand. Usually, it's faster and more flexible to sketch out basic letterforms on paper. Choose a few key letters to design and start getting creative.
Only now should you move to the computer. You can convert some of your sketches to vector in Illustrator by tracing them if you want, so you can use them later. But if you really want to design a complete font, you should use a dedicated program like Fontlab studio, Glyphs, Robofont or FontForge. Most offer a free trial, FontForge is free. These programs are extensive, and have quite the learning curve, so be prepared to get your hands dirty.
Test early and test often. While creating your letterforms, pay extra attention to common words and ligatures. Test different spacings, sizes, kernings, etc. Try printing your typeface, sometimes this will reveal things just not obvious on a screen.
- Beware of the rabbit hole
First and foremost, designing a typeface should be fun. It's quite a steep learning curve and the task can be quite daunting. One article will always lead to the next, and before you know it, you've read three books on typeface design, and haven't created a single letter. As said before, don't be afraid to get your hands dirty. Create a few very basic typefaces first, and step up your game with every one that you make. Revisit old ones once you get more experience and refine them.