Hey Joshua, hope your enjoying riding so far. Being new is one of the best times because everything is such a new experience.
Did you take the MSF? If not, you really should.
I agree with the other posters about shifting at higher rpms but ya might not want to just jump in head first shifting at 13k revs, lol. And fyi... the lil ninja isn't known to have the smoothest of 1st gear shifts, unless you're pretty smooth with the clutch. Some 250 riders hated it so much they installed a 15t front sprocket to give them a bit more range in 1st.
1st thing I would do is ensure you clutch cable and chain are adjusted to spec. I believe there is a how to thread floating around somewhere.
Next, get to an empty parking lot or clean, clear area and take some time to get intimate with your clutch lever and friction zone. During the MSF, the 2nd range exercise is the friction zone/duck walk. I know you can already ride but do this again. The point is to not only feel the friction zone, but also to start building some familiarity with where the friction zone is felt via the lever. The 3rd exercise specifically addresses your issues of shifting from 1st to 2nd and beyond. However, without having a firm grasp on the friction zone, the shifts can be jerky.
As a RiderCoach, I would recommend that you revisit those 2 exercises and continue to build your skill from there. Just like the throttle & brakes, the clutch is a finely tuned instrument. It's use can have a very subtle or a dramatic effect on what the bike feels like. The better you are at using the clutch, the smoother the bike can feel.
Lastly, the throttle and clutch have a close relationship. One has an effect on the other. There is a "sweet spot" on the throttle when shifting. This sweet spot is technically this; engine speed = road speed. With enough time spent riding, you will build your 6th sense being able to match the two without even thinking. Then poof... most of your shifts will be smooth.
Get out there! Practice and have fun.