Difficult to start and stalls at red light and stop signs.
 

Everything was fine a few weeks after i got my 2002 gz250 in march 2014. Its july now and my bike is giving me a hard time starting (i have to use the choke and try maybe a dozen times) and also idling while at a stop sign or red light. Multiple times Ive had to restart it at a red light. It doesn't go faster than 40 unless I get wind drift or Im going down a small hill. I changed the oil, battery, and put new gas in, and currently have the carburetor out and getting cleaned for tomorrow. I noticed on the right hand side of the carburetor there is a black tube coming out that leads into nothing. I was thinking this might be it. It comes out from the right side, but on the backer area.

I believe that is a overflo tube & remains open. ;) :)

Yeah, with a bend at the top and angling down, that's an overflow tube.

For good measure, we all recommend that people run some carb cleaner in every few or every other tank of fuel. The carb is essential on this little bike so you have to keep it clean.

Reading what you wrote, I'm going to suggest that you inspect the valve clearances. Hard starting is a tell-tale sign that they're not in spec. The carb probably needs cleaning anyway, and could possibly be the overall issue - but I'm trusting my gut here. Check valce clearances