"If you hear a new sound from your bike, stop and solve it immediately." Good advice from Bill Fridl, cyclist.
One of the great things about bikes is they are low tech and are not difficult to maintain or repair. Even if you have never used a spanner in your life you can easily acquire the skills you need to maintain and repair your bike while you are on tour.
The spares you need to take on a trip depends on where you are going, i.e. how close you are going to be to good bike shops, and how long you are going for.
For a tour of 2-3 weeks or so, in Europe or North America, where you are never going to be more than a day's pedal from a decent bike shop, I would restrict my spares to:
A couple of brake blocks
Some steel wire
A few cable ties
For a tour of a several months going into remote areas, e.g. Istanbul to Lhasa, the spares list would be:
Inner tube x3
Puncture outfit with lots of spare patches and glue.
Brake Blocks 4 pairs
Brake hanger (cantilever brakes)
Spokes - a few of each size
Front axle and cones
Rear axle and cones
Bearings (front and rear)
Derailleur jockey wheels
Plenty of cable ties
A good length or baler wire
Misc nuts, bolts and pipe clips
A roll of Duck tape
Small bottle of oil
Small amount of grease
If you are wondering what the cable ties, wire, pipe clips and Duck tape are about, they are for running repairs. If something breaks on tour, the chances are you'll be able to make some sort of repair with these.
Before you set out on a long trip, a good thing to do is take you bike almost completely to bits, clean and lubricate it, and put it back together again. This will ensure you have the know-how, confidence and tools to service it on tour.
For a long tour, where you may some days away from a bike shop, the jobs you should definitely be able to do include:
Mend a puncture (NB most people replace the tube then fix the puncture after the day's ride.
Replace and adjust brake blocks.
Adjust the gears.
Replace a brake or gear cable.
Replace a rear spoke - if you can do it on the rear, a front one is a doddle.
Re-true a wheel.
Strip down and refit wheel bearings.
Repair and refit a chain.
All the above can be easily learned from a book (try The Complete Book of Cycling by Eugene A Sloane) or by attending a few bike maintenance classes.
To do the jobs listed above, you'll need the right tools. Exactly which tools you need will depend on you bike, but here's an initial list:
Set of allen keys - check you've got one to fit every size of allen nut used on the bike - look out for the big one on the rear cassette and the tiny ones on derailleurs and gear shifters.
Small screw driver
Set of spanners to fit every nut on the bike - normally 2-3 small spanners, a headset spanner and an adjustable will do it.
Pair of cone spanners
Spoke key - make sure it fits your spokes
Chain whip (to work with the freewheel remover
Small wire brush for brushing the crud off your chain and gears.
It is well worth investing in a lightweight toolset to cover the above.