mig welding is the best way to weld brackets onto an axle. tig is a better option but I wouldnt suggest it unless you know how to and have a lot of confidence in your tig skills.
arc/stick welding would be a better choice if you were welding things together that were aout 1" thick. at that point is gets more complicated than just welding, you have to have a clean chamfered joint, then do a tig root pass or 2 then go over that with the arc/stick to fill the rest of it up.
make sure there is bare metal and a nice chamfered valley to fill, the joint/weld will turn out very clean and strong.