NOTE TRANSMISSION TIMES between this post and justjeff's above; we apparently were typing simultaneously!
I do not attempt to contradict justjeff's post at all; here's the tool list I can think of:
* Pliers (to remove axle Cotter pins)
* Wrenches/sockets (to remove axle nuts)
* Lift (or field jack or log or rock or whatever, to hold wheel off ground; or . . . lay her over!)
* Valve core tool (to remove valve core to deflate tube)
* 12 mm wrench or socket (to remove valve stem nut)
* Kickstand (or huge C-clamp, or boot heel, or automobile tire to break tire bead on rim)
* Tire irons/spoons (Two minimum; three more ideal, to pry/cam tire on and off rim)
* Bead Buddy (Motion Pro device, comes in field and shop models, keeps bead on rim when spooning; handy, but not essential)
* Lubricant (e.g., vial of dishwashing soap, to lubricate rim for seating)
* Talc (baby powder in a baggie to lubricate tube)
*Valve stem "snake" (not many have this; threads into valve stem to "spear" it into rim; not essential, but handy for the task)
* Spare tube (at least, for front tire; can be jammed into rear in emergency)
* Tube patch kit (patches and adhesive to repair tube, if you want to, or if you puncture your new tube upon installation)
* Inflation means (electric compressor and connections, CO2 tire inflation kit, hand (or foot) pump)
* Rim protectors (Optional, sheets of plastic cut from bottle (e.g., bleach, detergent) sides to prevent scratching rims with tire tools)
You can get by with less; improvise substitutes; but . . . with this collection, I think you'll have adequate tools to fix a flat or to change a tire.
More important: Your own knowledge and skill in HOW to perform the task, IMHO.
Seating the bead can be a challenge; a function achieved by over-inflating the tire after its mounted, 'til the bead pops correctly into its position on the rim; you can't use too much lubricant here, and may have to lube/inflate/deflate/repeat to make it happen. Windex is a convenient bead lube at home, but . . . problematical to carry in the field; dishwashing soap more practical on the trail, IMHO.
MANY techniques may exist for tire-changing; you may develop your own. Here's a video of Dunlop's technique (forget about RIM LOCKS, you ain't got 'em); he spears the rim with the valve stem easily, didn't notice the guy baby-powdering the tube:
Oh, yes; "Hope this helps!"