You want a standard 12mm cone spanner for the "inner" of the two nuts. If you really don't want to get one, fine nosed pliers can be used to follow Shaun's advice above (but the cone spanner is much better). As for how tight: you want the calipers to spring open easily (when there is no cable in), but not to have any play in them (no backwards and forwards movement when you apply the brakes and try to rock the bike backwards and forwards). I also use the "overtighten and slacken off" approach: tighten with the outer nut, slacken with the inner (holding the outer still as well).
If your version are like the ones here:
http://velobase.com/ViewComponent.aspx? ... a4831605f5
then you are (relatively speaking) in luck. There is a small hex fitting at the end of the spindle that you use to centre the brakes. Not sure what size it is off the top of my head, but about 4mm IIRC. You can use a small socket wrench (the black plastic sleeve just slides off if necessary). Care is needed, as they are easily stripped. I've not seen square ended ones, but that doesn't mean they don't exist...
I always smear a little light grease on the spindle, washers and spring when reassembling.