Zen and the Art of Cycle Maintenance
You need to keep your bike safe and roadworthy, and you can either learn to do this yourself (the best option) or pay someone else to do it for you. You won't need many tools for the everyday tasks and you can always get a bike shop to fix anything you don't feel confident about.
It is a really good idea to give your bike a thorough overhaul once a year (like a car's MOT), perhaps in the autumn before the winter weather sets in. Bike shops will do this for around £25 plus the cost of any parts.
Note that Portswood Cycles have a van parked next to the Student Union bike shed every Tuesday in term-time. You can leave your bike with them in the morning and they'll usually have it fixed by the time you need to ride it home. They can often fix minor problems 'on the spot' if required.
The 'all in one' tool sets from bike shops include most of the tools you need, but separate tools are generally more convenient and easier to use.
You need to keep these pumped up as squishy tyres make it harder to cycle and are more likely to get a puncture. You shouldn't be able to squash the tyre at all with your thumbs.
Equipment: a cycle pump, preferably one you can stick in a bag if you go out for a cycle in the countryside. Make sure you get one that matches the valves on your tyres - there are two basic sorts.
The best policy is to avoid having them by keeping your tyres pumped up, avoiding riding over broken glass and buying tyres with kevlar linings (more expensive but worth every penny). However, sooner or later you will get one, and this may be the point at which you get a bike shop to fix it for you. You can learn to do it yourself, but it is a deal of effort to turn the bike upside down, slacken the brakes, remove the wheel (especially the back one), remove the tyre, check to see if you can find the glass/nail that caused the flat, find the puncture and fix it (or buy a new inner tube) and reassemble the whole thing. With practice and the right tools it can take 20 minutes, but you may spend half a day cursing your way through this process as you find that your fix didn't work and the tyre still goes flat.
See bikewebsite.com (and subsequent pages) for instructions, or just search Google for "How to repair punctures" to find relevant YouTube videos etc.
You need to keep them adjusted so that the wheels do not rotate when you apply the brakes hard and try to push the bike. Replace the brake pads before they wear out completely, as that will ruin your wheel rims.
Keep an eye on brake cables and replace them if they are at all rusty or frayed.
See bikewebsite.com for instructions, or just search Google for "How to adjust bike brakes" to find relevant YouTube videos etc.
Equipment: a pair of pliers, allen keys or small spanners.
Gears and Chain
You need to keep these lubricated - it will make cycling easier and prevent damage to these expensive-to-replace parts. I recommend using WD-40 as it is easy to apply and doesn't collect grit like oil does. You do need to apply it fairly regularly though - perhaps once a month. Use the applicator tube that comes with cans of WD-40 and avoid getting any on your wheel rims, otherwise your brakes don't work! Try using an old T-shirt to stop the WD-40 dripping on the rims.
Equipment: can of WD-40, rag to protect rims.
The only other thing is to fix loose bits as soon as you notice them rattling or wobbling - for example if the pedals start making a click or squeak every time they go round. If you leave it, the chances are it will get worse quickly and cost more to fix.