From what I understand the most important thing to do when bitten by a rattlesnake is to get to a hospital. I often go on hikes where I am a few hours walk from civilization at the midpoint of my hike.
If I am bitten, should I attempt to run back to my car in the hopes of getting to a hospital faster, or should I just try to calmly walk back?
My concern with running is that it might spread the venom faster because my blood would be pumping, but I don't know if that's really how it works.
The concern is valid; your basic plan should be to stay calm and not increase your heart rate. If other options for evacuation are not available, slowly walking yourself out is the best remaining option.
The bitten area can swell quite a bit, so remove nearby jewelry or clothing to prevent them from turning into tourniquets. If you have a pen you can use it to mark the extent of the swelling on your skin along with the time; doing this at regular intervals will allow you or doctors to monitor how fast it's swelling. (The easiest way of doing this may be by measuring the limb circumference at the bite, and at a few places further up, and recording those numbers.)
For treatment, wash the bite with soap/water, and splint it just below heart level. When bandaging it remember to not wrap around the limb in case of swelling. If you happened to get a good look at the snake, try to remember it as that will help with identification, but don't go searching for it. Finally, don't use tourniquets, ice, incisions, sucking, et cetera.
References: My answer is largely derived from Buck Tilton's Wilderness First Responder text. One might also review Medscape's Rattlesnake Envenomation Treatment & Management, but in general the treatment recommendations for pit vipers are consistent across current sources.
First, to directly answer your question: no, don't run. Running increases your heart rate, which increases how fast the venom will be spread through your body. It also increases the likelihood of another injury such as a twisted ankle, which would further delay treatment.
Your first priority is to distance yourself from the snake and any of its friends. Walk slowly (don't run) back the way you came. The reason you go back the way you came is because you can be more certain there aren't any snakes behind you, but there could be more snakes in front of you. If you can identify the snake before leaving, do so, but do not get any closer to the snake while trying to identify it. There's no need to capture or kill the snake.
The next thing you should do is find somewhere safe to sit down. Snake venom can cause loss of consciousness, and you don't want to be walking (let alone running) should that happen. Taking a break to sit down, clean out the wound, and come up with an evacuation plan will help ensure that you get to help faster than if you just start booking it for the trailhead.
To clean the wound, wash it out using clean (drinking-safe) water. If it's still bleeding, apply direct pressure until the bleeding stops.
My wilderness first aid book recommends then doing the following:
Apply lymphatic compression wrap with 6" ACE wrap, distal to proximal, then proximal to to distal; leave in place during evacuation. Monitor pulse and CSM.
In English, that means if you have an ACE wrap (like this) you should wrap it around the bitten extremity, starting further away from the center of the body (distal) and moving towards the center of the body (proximal), then back. The purpose is to apply a small amount of pressure to slightly slow flow through the lymph nodes as this is one way the venom can move through the body.
Checking CSMs means checking the circulation, sensation, and movement below the location of the bite. You want to ensure that the wrap doesn't cut off circulation, and should monitor whether the patient's ability to feel and move their fingers/toes is decreasing over time. You may need to remove and adjust the wrap if swelling occurs, this is why it's important to check CSMs every 10-15 minutes.
You can also optionally take antihistamines (like benadryl) to slow the body's reaction to the venom. This will not prevent tissue damage, but can prevent an anaphylaxic reaction.
Once the wound is cleaned and optionally wrapped, you can walk out, but walk slowly to avoid further injuries. Make sure to take frequent breaks to check if swelling has progressed and to monitor CSMs.
Some things to keep in mind about snake bites:
