I agree with Kookaburra1701 that the most important tool in an emergency is between your ears. Take a First Aid and CPR or First Responder course from a certified educational organization. I also agree that a few pair of nitrile or vinyl gloves and some basic wound care supplies are all that is needed for citizen first response. I too am a paramedic and the kit I carry contains gloves, a 10-pack of 4x4 gauze dressings, a roll of gauze bandage, a roll of Coban (stretch gauze that has been coated with a substance to make it stick to itself but still be removable/repositionable), a small roll of 1" athletic tape, a CPR micro shield, a small mylar space blanket, and a pair of bandage scissors. The entire kit fits in a small triangle bag.

- If you are going wilderness trekking where help could be hours away, you should consider a more comprehensive trauma kit, but this should be adequate for initial response to the majority of emergencies likely to be encountered by a cyclist. If you take a good citizen first aid, first response, or wilderness first response course you will be taught how to improvise what you can't carry in a reasonably sized kit. You can never carry everything you could possibly need so learn to prioritize.

- Direct pressure is the first line of bleeding control. Use some kind of reasonably clean dressing (a shirt or bandana will do in a pinch) or even just your gloved hand to put pressure over the bleeding wound. A first responder or wilderness medicine course will teach you about arterial pressure points, which are also useful for controlling bleeding on the arms or legs. Many bleeds can be controlled with a properly applied dressing and bandage. A tourniquet should not be used if bleeding control can be accomplished by one of the other means, but is viable when a patient has a severe bleed that cannot be controlled by other means, when there is severe bleeding from multiple sites, or when you need your hands for other priorities like keeping their airway open. You do need to learn to properly apply a tourniquet so again, a first aid course is a good idea.

- Snake bite kits are of dubious value. Take one of the courses mentioned previously to learn the currently accepted method of treating snake bites. Whatever you do, don't pull out your pen knife, cut Xs over the bite marks and try to suck out the poison. This is folklore that is best left to black and white Western movies. Honestly, your chances of being bitten by a snake while cycling, even offroad in the desert, are right up there with being struck by lightning or winning the lottery. Just watch where you put your feet and hands and if you see a snake, leave it alone, North American venomous species aren't aggressive but will strike if they feel threatened. Outside of the US there are some species that are actively aggressive, so know the snakes native to your area.

- Don't move a seriously injured person unless absolutely necessary to protect their safety or provide treatment. Again, a proper course will teach you the lowest risk methods of moving or positioning a person. Assure that the person can breathe adequately, immobilize any injured area including the neck or back (have them lie still and gently hold them in position), and control any serious bleeding. Contact 911 with your location and details of the incident and the victim's condition. Stay on the line unless the dispatcher tells you to hang up.