This is quite a big question that covers a lot of ground.

The most obvious piece of good advice is, if a fire is burning in an area, stay away from it! Distance and more distance is the safest option. But that option is not always immediately available if the fire started near you after you are on the trail, so probably the best way to get started is to describe a few factors that affect fires and how they behave.

Forest fires move more rapidly uphill and more slowly downhill. They spread more rapidly the stronger the wind is blowing. They spread more rapidly in low humidity than high humidity.

Fire behavior depends a lot on the available fuel, how dry it is, how dense it is, whether it is low to the ground like grass and brush or high up like the crowns of trees. Generally speaking, fires move faster with low dense dry fuel. Grass fires can move miles in a day, while forest fires are usually slower moving, especially to begin with.

Bigger fires can 'jump' ahead of their front edges by sending burning material aloft which lands some distance away from the main fire. Low fires have trouble moving across watercourses, where fuel is wetter and the water creates a fuel discontinuity.

If you suddenly discover there is a fire in your near vicinity, your most important survival tool will be your brain and making good decisions, so staying calm and thinking clearly is extremely important. This will be more difficult if it is night time, but keep calm and collect information with your entire mind at work. Your basic task will be to get a safe distance between you and the fire.

Try to figure out what kind of fire you are facing, how close is the leading edge, which direction it is likely to be moving (test the wind), how fast it might move (think about fuel load, wind speed, humidity, terrain) and look at your map to see what bail out routes you have. Trails and roads are much easier and faster to follow than a x-country route! Incidentally, roads make good fire breaks.

Depending on what you decide you are facing, decide what direction you will travel, how fast you think you'll need to go, and how far. Don't panic, but don't waste time.

This info is just a few basic tips and hints. Fire fighters could tell you stories all day long to illustrate these brief hints and clues, and much, much more.

Later additions: Brian Le is correct that smelling smoke is not a good indicator of proximity. I have stood on top of a mountain and smelled smoke from an enormous fire that was burning almost 400 km away! Seeing a pillar of smoke is a much more useful indicator of location. If you see or hear flames, the fire is exceedingly close and I'd recommend moving very briskly!

Brian is also correct that having a fire start right near you is a rare occurrence. It has happened to me, but I hiked for 50 years before the first time it happened.