Having been a Paramedic/Firefighter/USCG veteran and in EMS/Public safety for over 20 years I have a couple of thoughts on the subject. A Basic First Aid course as well as CPR training are good skills to have not only for chasing but also in general every day life. Both classes are very easy and inexpensive to actually get certified in. If you want to go a step up in First Aid EMT-B would be3 the next level with a couple hundred hours of training with recertification required every 2 years ( at least here in MN) and you need to be working in the field to get re-certified so First-aid classes would be easier to keep current on.
Having been Incident Command/lead medic on many MCI/large incidents I have come to appreciate help that just general people with some basic training can provide. Simple things such as basic bleeding control/splinting, stabilization skills can be immensely helpful as most of these skills are basic and can free up people with more skills to help those more in need. Basic Triage/START/RPM knowledge can be helpful to know the basics of and I am sure that this info can be found online(I have not taken the time to look).
A basic first aid kit with bandages, dressings, gloves, Kotex (these rock for laypeople and inexpensive) is easy and cheap to carry. You can easily add these to a "kit" you carry for yourself chasing with items as antacids, ibuprofen, aspirin, etc. You can also carry as much as you feel comfortable with using CORRECTLY and safely. I myself carry a pretty advanced/complete first aid kit, but then again look at my history of knowledge of using stuff like this working busy Major 911 areas for years as well as rural 2-3 hour drive time to hospital 911. As far as a Tourniquet goes I would NOT apply one unless you KNOW what you are doing as much worse damage can occur and if you apply one w/o having proper knowledge may be a bad thing as far as the Good Samaritan laws go. I also would advise against giving Aspirin to people complaining of chest pain because it is technically a medication and since you are not a Dr or someone working under one you may get in trouble for giving it to someone else.
First and foremost as some people have mentioned is that you yourself be safe. You can NOT help someone else if you become hurt/injured yourself and if you do there is just more strain on the system.
I apologize if this seems to jump around but that may be my ADHD running my brain/fingers. Having basic skills and knowing how to use them would be IMO the best things to have to help in scenarios like these.