Below is a pic of some of the stuff you might want to carry with you...

For the typical tire repair...you'll need the two repair tools at bottom left....one is a REAMER to clean out the puncture hole in the tire AFTER the nail or whatever has been removed from the tire using needle nosed pliers. Reaming also opens the hole up some.
The other tool is the repair pug INSERTION tool which has a split in it where you place one of the fiber/rubber strips shown at the bottom...in the split area with about 1/2 left on either end.
You then put some rubber cement on the reamer tool and try and get some into the puncture hole...then clean off the tool.
Then you cover the mounted plug with the same rubber cement covering all surfaces well.
Then comes the FUN PART. You next need to force the plug into the tire. You WILL want to take note of the direction that the nail or whatever went into the tire...and follow this exactly as you insert the plug. Sometimes this can take some serious force...so you don't want the cheapest tools you can find. You will need to follow the instructions that came with the tool...but generally...you pull them straight out after you've inserted the plug far enough that the open end or split is INSIDE the tire. Once the tool is out...you then cut off the excess on the outside...maybe leaving enough to get a hold of with your needle nose pliers...in case you'd want to remove the plug later.
My experiences with this type of tire repair have been mixed....some times it requires a second plug before you can get past a slow leak...where the tire will still slowly leak down...a little spit or soapy water will show up this type of leak when placed where the plug surfaces. 100% of the time...a second plug done in the same way...fixes it for good. You can either pull the plug out with needle nose or sidecut pliers...OR just push it inside the tire using the reamer tool.
Many times you can fix the tire while it's still on the car.
As far as what AREA of the tire you can repair...usually "they" say only the tread area. I have repaired tires maybe 1/2 inch to an inch above the tread area and have had them "hold"....and had an old dude who runs a tire repair place say he repairs sidewalls this way...but if you are up in the area where the tire sidewall flexes a lot I'm not so sure...plus the sidewall has much less fabric cord in it.
You ALWAYS want to use the cement...it also acts as a lube and you will find it very difficult to get the plug in without using it.
A problem with the rubber cement is that once opened...it can dry out and be useless when you need it. I try to keep the tube wrapped up in about 3 ziplock sandwich bags to slow the solvent evaporation when stored.
The next issue is...now that the tire is plugged...how do I get some air in it?
I always carry a 12V air compressor to re-inflate a repaired or low tire. These can cost from $10 to $20 and even more. The cheapo $10 compressors are surprisingly durable...but I usually need to disassemble then before using to lube the shafts and gears and maybe tighten screws and do some "swaging" here and there to keep bushings and so forth in place. Don't lube the piston though...maybe light oil at the most. Recently bought a Ryobi 12V compressor that should last a bit longer and puts out a little more air....but usually the better they are the bigger they are. Probably don't want to use these in the rain...it will crap out the motors.
You can also buy the "spare in a can" type inflators that use latex and a propellant to inflate the tire...but they usually don't fully inflate a tire.
Also handy is a tire valve tool (not pictured) that allows you to to RESEAT the valve core inside the valve stem in case it is causing a slow leak...you just back it out a little and retighten it...but not too much....sometimes the onset of colder weather can cause these to start leaking.
And a small dial type pressure gauge is handy to know when you have the tire inflated correctly.
How to find that elusive SLOW LEAK:
Sometimes whatever punctured the tire will not still be in the tire...or you might have a bead leak at the wheel...or a leak at the valve core or stem.
Usually in these cases you'll definitely need to remove the tire/wheel. I usually use a garden hose to thoroughly wet the tire down and get any debris off of it. Then I mix up a quart or more of soapy water...you want soapy water (reduced surface tension), but you don't want sudsy water.
You basically pour or dribble the soap water starting on the tread looking for air bubbles (you'll need air in the tire)...need to be real careful in looking...because the leak in sometimes not very fast or obvious.
If you don't find anything on the tread...you can then check both bead areas by laying the tire down on it's side and pooling the soap water in this area. Then check the valve stem where it inserts in the tire...and remove the valve cap and check the valve core for a leak.
This can take PATIENCE and you need to look closely.
note: feel free to add any additional bits of wisdom
