The standard kit carried by any member of an away team will vary from ship to ship, environment to environment, and person to person. It is highly situational but, in general here is what they will carry:
Everyone- Standard duty uniform.
- Commbadge.
- Type 2 Phaser (if necessary for the mission. Carried by security officers on shift.)
Department Specific- Medical officers will often carry a medical tricorder as well as first aid supplies.
- Science officers on away team will often carry a more generalized tricorder on away missions.
- If combat is expected, security officers will carry Type 3 Phaser rifles, though other officers may also take these.
- Engineering officers might bring some tools, possibly a specialized tricorder or scanner of some sort as well as a hyperspanner.
- Operations officers often carry data PADDs and/or general tricorders.
The answer to that question is- whatever it is dramatically appropriate to have on you at the time. While it's unlikely to that you'd have a giant sword that you failed to mention before or the EXACT right key to a jail cell- it's reasonable that a security officer might have taken a small bit of explosives if it was needed or a medical officer might have a dermal regenerator on hand. No one is going to give you a hard time if you pull a ration out of nowhere to feed a hungry orphan.