How can I stay safe when chasing a storm?

1. Educate yourself. Read all you can on the science of storms, and don't limit yourself to articles in popular magazines. Look through some of the articles in the Stormtrack technical library. Chuck Doswell is a researcher who has been exceptionally diligent about posting the best information online; check out his papers at his professional and personal site. Also ask questions here on Stormtrack. Watch storm chase videos sold by other chasers to see, uncut and raw, how they approach a storm. The stuff shown on TV tends to do a poor job at this, since the editors and producers are not storm chasers.

2. Keep aware of the storm at all times. Don't turn your back to what is overhead or behind you. Soak in the storm as you watch wall cloud areas, lowerings, and tornadoes. You have to always have the big picture to predict what the storm is doing and how it is changing, and this will key you in on unexpected developments.

3. Always have escape routes. Do not go down dead-end roads. It's best to be aware of (and close to) roads that are perpendicular to the storm's path so you can make that quick escape if the core collapses or a wedge widens just to your west.

4. Stay in your car. Lightning is unpredictable, and more and more incidents are making news of chasers getting zapped by side flashes (fortunately no direct hits yet). When under the updraft base, you don't really need to be outside. Use turnouts to orient your car for that optimum shot, and use interior tripods such as fixed monopods and window clamps when possible. It will keep your electronics safe from that rogue raindrop, too.

5. Keep off of roadways! The media likes to air footage of chasers parking with the rear end hanging in the roadway, and unfortunately this spawns similar behavior by other new chasers. It's a major safety hazard and can result in a collision from distracted drivers and speeding motorists. Parking on roadways is not accepted and is scorned heavily by chase veterans. Get out of the lanes and pull over as far as you can. Even on remote stretches of highway it's second nature for a skilled chaser to find dirt road intersections, gravelled paths to cattle gates, and widened shoulders near bridges. These are safe. One hint is to look for telephone poles leading away from the road; this often suggests a turnout.

6. Don't chase at night. Though some experienced chasers do it, ask around. You'll find that a surprising proportion are loathe to go after a storm when it's dark. Awareness of the storm's danger is seriously degraded given a rain-splotched windshield and poor illumination. Lightning photos, especially anvil displays, are usually not a problem, as you can usually keep a safe distance from the storm and not get near a wall cloud or in a core. Sometimes the best lightning displays are found behind the storm, and you can follow it without a problem if you watch out for downed lines and debris.

7. Drive defensively! Your biggest danger is other drivers. Underneath an updraft this will consist of distracted and scared motorists, local teenagers out looking for a cheap thrill, and storm chasers -- a handful of whom have little regard for safety. This is especially true if you're chasing anywhere in central Oklahoma, where chasers and "inspired locals" are especially active. Always be prepared for the worst.

8. Keep your gas tank full. If you are chasing what turns out to be a big outbreak, power grids will almost certainly get knocked out and you'll be unable to find fuel. Be prepared for power to go out anytime storms are around, lest you enjoy camping out in your vehicle overnight.

9. Put a little maintenance into your vehicle. This is common sense stuff, but if you get stuck out on the back roads of western Oklahoma, you will truly be up a creek. You also don't want a blowout at 70 mph on a narrow farm-to-market road. Get familiar with your engine so you can recognize what is normal. Learn how to change belts (carry a spare), deal with an overheat, and change tires quickly. Take special care of your tires and test your jack to make sure it will perform correctly when you need it. Tires are your lifeline home!

10. Avoid chasing in cities and on busy freeways. Chasing in cities can be done, but it's often frustrating. Traffic jams are frequent as rush hour coincides with the peak of convection. Toss in rain, storm cores, and people blocking traffic to park under bridges, and it can be a mess. This may pin you down with no escape route and put you in the path of danger. Also impromptu police roadblocks are common in metro areas which may thwart your pursuit of any photogenic tornado.

11. Keep away from those Allsup's burritos. You've been warned.
If worst comes to worst and you find yourself requiring immediate shelter from a tornado:

Best resort — a building very close by would be the best possible shelter, because you ideally want to shelter yourself from flying debris for the duration of the tornado — when winds may be coming at you from every bearing at one time or another; in that case, something like a single freestanding wall won't do the trick.

Worst resort — (a.) An overpass. This will only be of moderate protection if there is a distinct recess that you can shelter in, and/or a handhold (e.g. girders on the underside of the bridge above). Any other kind will not help. An added danger is that there may be crowds around an overpass, from either people seeking shelter from the tornado, or hail preceeding it. In that case, it maybe difficult to find space underneath, and all the vehicles around can add to the flying debris.
(b.) Your car. Cars may be designed for some safety in rollovers, but that's not counting on the car being rolled 100+ feet (which a minimal F2 could handle), or becoming airborne for some distance at some speed. And even if you are wearing a seatbelt, it wouldn't take more than a sudden movement of the vehicle to jolt you forward into the steering wheel, or dashboard — sideways into the window — up into the roof — to cause debilitating, even fatal brain trauma; concussion at the very least. It also wouldn't take much to break the windows, either by impact of wind or flying debris, leaving you exposed to both.

Last resort — a ditch. Be wary of fast-flowing water in the ditch due to the rain from the storm. Lie as flat as possible and cover your head with your arms. Grab onto the base of a small tree/shrub if one is nearby. You are at risk from flying debris here, but you would most likely suffer less injuries this way than being any higher up.
I have to emphasize Tim's #2. Be aware of what's happening around you. Because I was keyed into a tornado dropping about 2 miles in front of me, I wasn't paying attention to what was happening around me. I had a small (F0) tornado drop within 200 yards of me! I just happened to look over my shoulder and saw the dust swirl. Then I looked up at the clouds. There was no doubt of what was happening. My only consolation was that the three other vehicles watching this weren't aware of the dilemna either!

Pretty dumb, eh? I wasn't going to stick around to find out if it was going to intensify or not (it died out a few moments later not running 100 yards). My head is now on a swivel when I'm in close to a storm.
I would have to agree with Tim and John that situational awareness is key. I hear this every year in spotter-training and I am so happy to hear it: "Don't get tunnel vision!". If you see something that looks interested you may spend your time watching it while you don't realize something potentially dangerous going on behind you.