I am operating under the fact that you know what Syndra's abilities do and you know how they work with each other. I recommend you read up on her abilities and how they combo.
ALSO! A lot of the information that would be in a normal guide is in the notes tab that you need to hover over.

Your goal is to harass as much as you can with Q. Similar play style to Karthus for farming. At level two I would try to go in for a kill with a Q-W-Q-Ignite combo, of course the ignite depends on how low they are but if you can harass with Q level 1 you can probably kill. At 3 you can try for a kill if you couldn't at level 2 and he stayed in lane. From here on just farm until level 6 where you just Q until they are at about 60%-70% (while still farming) Once you have done that Q-E-Q-W-Ult. That should net you a kill, now unless you need around 50 gold don't stay in lane because there is the chance that you die via Jungler gank. From now on just try to harass and farm as well as you can. Remember, a kill = ~15 minions of gold. Use that to judge whether you want to fight or just farm.

Basically you want to go for picks. You should be able to solo kill most champions if you have a few kills. If you arent ahead just stay back and farm whatever you are able to. Farming is super important if you want to stay relevant and you should be able to farm as your wave clear is exceptional.