There is a difference between a first aid kit and a medical kit.
I do not know the extent of your medical knowledge, but am assuming that you need basic info/should not really be self medicating else you would not have asked this. Any medicines/treatment - you should find a doctor unless you are very sure of what you are doing. So I will stick to an EMERGENCY kit - a small one that should see you through till you get proper care. You can easily buy all of it separately.
The main emergency - cuts/lacerations (or worse) if you go rubber side up. Sunburn. Insect bites. Most other stuff is usually not an emergency, please seek trained advice. As such, I will not go into stuff for stomach upsets and the like.
a) Any dispersible OTC analgesic anti inflammatory that you are not allergic to - something with a 8 hr duration of action or more preferred (eg nimesulide/etoricoxib/diclofenac potassium). Dispersible - acts faster, not to be taken on empty stomach, and not in the presence of pre existing gastric irritation
b) Gauze bandage rolls. Handy for holding pressure packs in place to immobilizing injured limbs
c) Cotton roll +sterile non adherent dressings+ sterile gauze pack
d) Self adhesive patches, various sizes (eg band aid), and a roll of micropore adhesive bandage.
e) Local anaesthetic solution lignocaine 2% with 1:200000 - 1:80000 adrenaline (for injection). Not sure whether you can get it without a prescription. Applying some on a scrape (soak a piece of sterile gauze in it and allow to stay on wound) makes subsequent cleaning much less uncomfortable. Also, the adrenalin promotes haemostasis, initially.
f) bottle of clean drinking water (use it to wash contaminated wounds and get rid of road debris). This is preferred rather than using a harsh antiseptic soln which will kill a lot of healthy cells and DELAY healing. Plentiful in supply, too!
g) Povidone iodine ointment - apply on emergency dressing. Liquid can be used too. Be sure that you are not allergic to halogens! Or any other such soln/ointment.
h) Sharp scissor, pointed tweezer, pkt large safety pins.
i) Calamine lotion.
j) Any proprietary spray/patch for relief from insect bites.
k) Small bottle of alcohol based hand wash.
l) Nitrile gloves, sterile preferred. A lot tougher than latex. Get two pairs. Getting somebody else's blood all over your hands is not something you will enjoy. Even if they are not sterile (examination gloves), they will protect you.
m) A packet of ORT salts (actually, if really hot, you should be drinking a solution of this rather than just having vast quantities of plain water - cramp inducing). Useful for severe diarrhoea, to stay hydrated.
Other medicines would be part of a medical kit, as opposed to a first aid kit for biking. Support the local GP!
And last but not the least:
LEARN HOW TO USE IT!