Essential tools (if you own the house/flat especially) to my mind would have to include
Decent screwdrivers - inter-changable bits, a set of normal screwdrivers, and a set of "jewelers" type small/precision ones.
Pliers/wire cutters
Hammer
Tape Measure
Spirit levels - a small and a reasonable size one, if you're putting up a lot of shelves something like the Black & Decker self levelling laser levels are very handy for getting a long level line (much much easier than using normal levels, especially turning corners such as when putting up tiles or lots of shelving).
Half decent drill
Spanner set - 2 off (one cheapy, one good set - chances are you'll end up needing more than one of some spanner sizes, hence two sets) - you may end up getting 3 sets (imperial as well as the metric ones).
Socket set (same as spanners).
Adjustable spanners - small, medium and 2x large (you'll be glad of them when you do any plumbing).
Mole Grips
Plumbers wrench
Pipe cutter
Tyre Irons
Hack Saw
Wood Saw (get a half decent one, Stanley do some very good ones)
Gas torch + heat proof cloth (saves setting fire to the wallpaper)
Allen Keys
Radiator keys (you should really get 2+, keep one on your keys, and one in the tool box where you'll never be able to find it when you need it)
Tape measures - ideally at least one small one, and a couple of longer ones.
A couple of tool boxes to put everything in.
Consumables you should keep in stock if you own your own house/flat.
Screw Driver bits - get a decent set, don't be surprised if it costs a couple of pounds a bit to get some good ones (they last longer, and are much less likely to cam out of screw heads).
Drill bits - get a decent set of both masonry and wood/general bits, they make a lot of difference when drilling.
Wire wool
Sandpaper (just a few of each grade unless you're doing some work)
Spare blades for any saws that take them.
A few suitable connectors for your plumbing - if copper get a couple of each of straight connections, right angle and T joints for example.
Suitable pipe for your plumbing (even just one length will be a great help at times).
Pack of assorted washers.
Fuses - about 3-5 of each of the common types (3/5/13 amp)
Electrical screws - most fixing seem to come with the shortest possible ones, so it's a great help to have a pack of 45, 55 and 70mm ones hanging around.
Pencils
Assorted screws/nails/pins
If you've got a bike or anything that uses low pressure tyres, a puncture repair kit.
Some 3, and 5 amp screw block strips (most light fittings seem to come under the assumption your lighting circuits aren't the ring type we have in the UK, and so only come with a connector block for 3 wires, rather than 4 for example).
PTFE tape + Fernox LSJ (I think it is), both are god sends when plumbing
Safety/assorted
Goggles
Ear protectors
Rubber gloves
old sheets/dust sheets
We've got multiples of pretty much everything mentioned above (and i've not even gone into the power tools/more unusual stuff*), and it's extremely handy being able to pop into the garage at a moments notice and pick out the washer needed to fix a dripping tap (or even a new fitting to replace a damaged one), or the screws to get that new light switch to actually fit**.
It's not so handy having the neighbours know that we've got a full set of drain rods, and getting roped into unblocking their drains when they've been flushing nappies down the loo :/ (hasn't happened for a while thankfully).
Two tips, if you ever need a tool to do a job get it, you might not use it again for years, but they do tend to pay for themselves, and if you see a tool you might need soon cheap, pick it up, they don't come to any harm sitting in the tool box and chances are you'll curse when you do need it and have to make a trip out specifically to buy it (usually whilst you've got the water turned off due to a leak...).
*My dad's never thrown a tool away and has things like sockets & spanners in 3 or 4 different types (from memory he's got metric, imperial, whitworth? and another long obsolete type that crops up from time to time).
**We've actually got several tool boxes filled with things like plumbing fittings, electrical spares, boxes of screws in pretty much every size and type from 12mm phillips to 150mm coach bolts.
