It doesn't jump from 2nd to neutral, it just doesn't go from 1st to 2nd all the time.

 

in order...

1 check oil level.
2. when was your last change?
2.5. clutch adjustment
3 clutch still good
4 fubared, take it easy and ride it till it goes out or rebuild.

 

Could be a problem with the clutch plates not disengaging as the tangs and ears can get 'burred' over causing drag and poor release. I'm not sure about the oil being a big issue unless it is filthy! Won't hurt to change it, but get yourself the Haynes manual for you bike and check how to dismantle the clutch basket and carefully check all the plates to see if they slide smoothly in the basket. Look for evidence of 'blueing' on the tangs, which means they have been stick and getting too hot. Poor clutch cable adjustment can cause it to drag, but as you seem to have it adusted correctly I'd go for the clutch plates. Fork selectors do wear as suggested, but change oil, if that doesn't help, check clutch plates and then fork selectors. Isn't life wonderful!

 

As people have said, check your oil. Use motorcycle oil, not car oil. Your clutch could be acting goofy if car oil was used. I would think your clutch adjustment is fine. If you are using the clutch the same way every time, then it must be working. To me, it sounds like this might be in your head. F2s are notorious for wanting to go into neutral from 1st to 2nd, and generally very clunky shifting. It sounds like you are new to the bike. Really focus on jamming the shifter up and down when you shift. See if you notice that "problem" going away. Does the bike ever want to keep creeping forward while in gear and the clutch lever pulled in at a stop?

If it is none of these things the above people mentioned, there is something going on with the transmission.

 

I just reread your post. When you say you have to re apply the clutch to get it to shift, does that mean when you let go of the lever the first time, the bike wants to free rev like it is in neutral? Or is your transmission still engaged? If it free revs until you pull the clutch lever in a second time to shift, it is almost certainly what highwaypatrol said about the clutch plate tangs and ears getting burred, making the plates stick.