I don't solder, although I have no problem soldering. I used to solder nearly all day when I diagnosed and repaired circuit boards at a factory. I have built kits that required soldering >64 sockets, each with 18 pins, to the circuit board. I still would rather not solder, and there is more to joining wires than soldering. Soldering is to prevent corrosion and eventual bad contact. You have to join the connection securely mechanically first. Afterward, you need to insulate the connection. Electrical tape will do. You can also use heat-shrink tubing.
One reason I don't solder electrical connections, as opposed to electronic, is that I am forever changing my mind about what I want, and I prefer something easily reversible. The other reason is you don't have to. Electricians generally don't solder. They use "wire connectors", also called "wire nuts," little conical plastic thingees with a spingy, screw insert inside, although the smallest ones often are completely plastic. All hardware stores have them in the electrical section. The small ones are cheap. You need about the smallest ones, for #20 wire probably. You put the two (or more) bare ends together side by side and twist the wire connector on the ends. If you do it right, the connection is completely covered and insulated. For insurance, in case you jerk on the wires hard, you can wrap the wires with electrical tape, stetching it tight for a couple of loops and then not for the last couple of loops.
If you need to strip insulation, they have a multipurpose tool ("electrical pliers") in the electrical section that looks something like pliers. Unfortunately the tiny fan wires probably are too small for the smallest stripping hole. There are also wire stippers that adjust to any size with a little sliding nut, but they are kind of hard to find. What electricians seem to do though (probably because it is fast) is to just catch the wire a little in the jaws of a wire cutter (diagonal pliers) so it starts a tear in the insulation, holding the wire in the crotch by pressing a little on the wire with one finger of their pliers hand, and jerk. That rips the insulation. Yeah I know it sounds kind of tricky. Another thing which works is to crush the insulation in some pliers. That damages the insulation on small wires, after which you can pull it away and cut it off. The good thing is it doesn't put a nick in the wires.