trick I use at work (for anything up to 18awg wire) is to wrap one turn of wire around the shank of a screw driver.... unspool more than enough wire for the job, make the wrap as close to the coil as is comfortable, then hold both the reel and screwdriver securely and pull the two away from each other... the single wrap puts just enough friction on the shank to let you slightly strain the wire, drawing out any kinks in it.. Just mind your fingers as you near the end of the wire... it can bite..!! Thicker the wire, the bigger the ouch...
Should add that trying to draw anything larger than 18awg takes some serious grunt... hold the screw driver in a vice if you can... goes without saying to be careful... you might need to draw the wire through a couple of times to get out all the kinks...
To fully straighten the wire after its been drawn, hold the reel securely, hold the other end of the wire with strong plyers and draw (pull the two away from each other) the two as tight as you can... you'll chew up the end couple of inches of wire in the process, but the rest should be nice and straight / kink free... cut to suit your needs and / or repeat as necessary...
Last point... if the wire doesn't take solder cleanly (tins in some places but not in others) chances are that the top coating of the wire has slight corrosion... a light rub through something like scotchbrite should cut through any oxide...