Adjustment is much easier than it first seems. The chain moves between gears simply because the gear mechs push it sideways until it does.
With the rear mech, you move the lever until the chain is in the gear you want and then fine tune it backwards or forwards until the mech is aligned under the selected gear and there is no clicking or scraping (of the chain being pushed into the next gear or rubbing on the cages of the mech). Pretty much the same with the front, except there is less fine tuning to do. The only tricky bit comes at the limits. At the smallest rear cog, the mech needs to be able to drag the chain into gear but not pull it so far over that it gets caught between the block (=freewheel/cassette/collection of sprockets) and the frame. The adjustment screw for this is normally captioned "H" (High) and usually on your right as you face the mech. Make the adjustment with the chain in the largest of the chainrings at the front, ideally whilst the bike is held securely on a stand that allows the wheels to spin. It can be done with the bike upside-down, but you tend to damage the brake cables. When you have the adjustment such that it will change from the second highest gear to the highest, ensure there is no excess slack in the cable when the right hand lever is full forward. If necessary, release the cable pinch bolt, pull the cable tight (there's a tool for this but you can pull it with your fingers or carefully with pliers - you don't want to crush the cable as it will fray/start breaking) and re-tighten the pinch bolt. Now we must do the other limit, and this one is done with the chain in the smallest chainring. Take great care not to allow the rear mech to throw the chain into the spokes; wind the adjustment screw over far enough and it will let you do this and you can really mash your wheel. Likewise, don't limit the throw so much that the chain can't climb up to the big sprocket. Take it slow and watch what the mech is doing.
Now the front mech. Note before you start that the 2 sides of the cage (the flat plates that push the chain between gears) are deliberately NOT parallel when viewed from above, but converge at some point ahead of the bike. Occasionally the mech gets knocked out of position and if this is the case with yours, take a picture and come back here. Anyway, assuming the mech is in the right place (pretty likely), stick the chain on the largest sprocket at the rear and the smallest sprocket at the front (the lowest gear. Adjust the low limit screw on the front mech (usually closest to the bike centreline) until there is no rubbing of the chain on the inner face of the cage . With the left hand lever full forward, take up any slack as you did at the rear. Change to the next lowest gear (2nd largest sprocket on the back) and then using the left hand lever, change to the large chainring. Adjust the other limit screw to ensure that the chain is not thrown off the ring toward the pedals when making this change. Change to progressively higher gears with the right lever and check that the front mech can push the chain up and down, again adjusting if the chain is being thrown. You're done. When riding, always avoid using the largest chainring/largest sprocket and similarly smallest/smallest as the angle it puts in the chain wears it out unduly. Probably less so with modern chains but still a factor worth bearing in mind.
(Incidentally, you will now appreciate why you should always back off the pedal pressure a tad whilst you're changing gear. Push the pedals hard enough and the mech simply can't push it out of the gear it is in).
That's a load of words but about 5 minutes when you're actually doing the job. Simpler and more reliable than modern "index" (click-click you're in gear) systems but evidently only suitable for a better class of cyclist
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OK, the new ones are faster but don't let on to youngsters that there isn't a highly-skilled black art to your friction system.
Just noticed that Hamster is a faster typist than I am and offers good advice on your chain that you should follow before adjusting the gears. Good luck and shout if anything doesn't make sense.