So I bought a totally junked 2002 TTR 250 for dirt cheap awhile ago and have been slowly rebuilding it. Upon installing the starter I got a little confused as to some wiring and where the breather hose is. In the picture below, you can see that I have the starter wire attached to the top of the starter motor, sandwiched between two washers with the rubber boot on top of it. Next to the starter motor is the clutch cable plate and under one of the screws is a wire. Is this wire supposed to go here? What is it? Also, in the guide to reassembling a motor on this website, there is a breather hose going into the crankcase somewhere, but I have no breather hose to be found (and no opening in the crankcase for it). How stupid am I being? I'm pretty new to all of this and mostly just bought this bike to learn some things on. Does anything else look messed up in the picture? Thanks!!
The wire you have under the clutch cable bracket is the battery earth, as Brian says. I think it goes directly to the negative terminal of the battery, and normally attaches to the rearmost starter motor retaining bolt. The solenoid will have one thick wire from the positive battery terminal and another thick one to the starter top terminal as you have correctly attached.
Crank case breathers are different on 90's metal tankers and 2000-on blueys: The older engines have a breather coming out of the crank case itself just behind the starter motor, the pipe goes to the cyclone filter below the air filter box. Blue tank bikes have this breather hole blanked off with a core-plug, (you can see it in your pic - next to the clutch cable bracket) and the engine breather comes from the top of the camshaft cover.