Someone - I think Aubrey - said in one of his blogs recently that you have to assume that techies who fix your stuff have only three tools - an FBH, a FBS and and FBW (the last letters standing for Hammer, Screwdriver and Wrench. Pretty much on the money I think!
Some great suggestions here. One thing I have found absolutely indispensible is a set of screwdriver bits - some of these come with socket spanners (wrenches??) as well. I'm not talking about the usual flat / philips / pozidriv bits, I am talking about the all-singing, all dancing kits with a hundred or more bits comprising the usual ones above plus hex (allen) key bits, torx, security torx, tri-wing, etc, etc - usually with a ratchet driver handle. If you're just making stuff you probably won't need them, but if you fix stuff (and who doesn't occasionally?) you certainly will need them sometimes. Not often, but when the time comes they will get you out of trouble. You will STILL need something else sometimes - see my story here
to see how essential it is to have the right tool at the right time. As it happens, the story is about a drill - also an essential tool that no-one else seems to have mentioned. I like to keep a small, cheap battery drill at my workbench for drilling holes, and a bigger, higher quality one for bigger jobs. I also have a PCB drill on my bench.
@Max: For the beginer starting out and on a budget there are a lot of things that can be used as cheaper alternatives to the professional kit.
Like the use of the good old bulldog (is there a different name for this in America) paper clip (many sizes) as a heat sink and mini clamp. Boot lace ferrules especially if using multi strand wire on your screw block shield, stops the stray strand from shorting out to another block.
However what alternative do people use for crimping
That is a pretty good, fairly complete basic list.
I have an Optivisor for magnification, but I sometimes need to resort to a stereo microscope.
I'm an RF guy. An Xacto knife is my constant companion, along with a stick with a small piece of copper glued to the end.
I also build with tubes (!). A small wire brush for cleaning terminals (whether on tube sockets or on 3/32" headphone connectors or any other tarnished but solderable surface) is required, not optional. Sometimes the file will work for this, but not usually. A small piece of sandpaper may be helpful here, too.
I would add a metal ruler or (preferably) inexpensive calipers.
For making holes...a spring-loaded center punch and drill. The punch can be very inexpensive. For making irregular holes, add a nibbler tool.
Concerning the wire stripper: many of these also can crimp solderless terminals. Make sure your wire stripper is so equipped, and solderless spade lug terminals, butt connectors, and some multipin connector terminals become usable.
That screwdriver set should be one of the sets with umpteen billion bits...because you WILL need esoteric bits sooner or later. I include sockets/nutdrivers here.
I would also add a small (6" or 8") adjustable wrench. Some things just need to be gripped from the side.
I like a cigarette lighter for quick jobs with heat-shrink tubing, or lighting my torch. For that matter, electric tape, epoxy, heat-shrink tubing and nylon tie-wraps are very useful for bonding things.
These are just a few off-the-cuff things. But I have basically all of these on my bench, and most of them in my pickup truck tool box.
Finally, for those just getting started...don't get TOO hung up on quality. We all have to start somewhere. When I was 9, I kept my tools (what few I had) in a Keds shoe box, and my workbench was a folding card table. My projects didn't know the difference, and it didn't keep anything from getting done.
I often find that just blowing by mouth is enough to get rid of enough solder, depending on the situation. I guess a blast of compressed air might work (NOT canned "air" as many of these could ignite or give off toxic fumes if they come in contact with hot surfaces).
Sorry to intrude, but I hope my messsage helps. Blowing a componenent out from the mother board can be difficult as the componenent maybe firmly routed. The usual practice is to weaken the solder and suck the component out. This appears better to me.