You don't need a syringe unless you get reservoir types (more advanced). Just start with plain cartos.
try Boge soft-caps. With a Riva 510 you can use either standard or the hotter low resistance ones. And I used their clear plastic drip tips in cartos - not to drip since the jucie would aim for the air channel - just as a cool easy-to-remove mouthpiece
Here - the Boge and/or the low resistance Boge
Joye 510 cartridges
and here is a small badly stocked selection of prefilled 510 cartos - I don't know if these are softcaps - may be harder to refill
Xpress 510 Cartomizers
And here is a video that shows how to fill a soft-cap carto as well as how to straighten the center air channel if it gets crooked. This video is filling a soft-cap KR808D-1 carto - similar with different threading. A small unbent paper clip stuck in the mouth hole just enough to pop the mouthpiece out is all that is needed. I hold a tissue on the threaded end in case it leaks. I hold it at an angle like in the video but if the stuffing is tight I rotate the carto to drip into different spots. You should be able to fit about 18-24 drops into a 1ml carto depending on the size of your bottle's drops.
When done I clean up the threaded end. I also tilt the open end down into a tissue and puff into the threaded end to clear the air channel. Then I use a folded tip of tissue to clean up any loose
juice pooled above the stuffing (else it will end up in your mouth or on the battery). Then cap. On a fresh fill of a dry blank, I then lay the carto on different sides over about 15 minutes to make sure every bit of stuffing has absorbed liquid (else you could singe the stuffing). I use 70pg/30vg juices - if you use thicker juices you may need to let it soak in longer. I do not use condom push methods to fill cartos.
YouTube - How-to guide to refilling KR808D-1 or Vapor King cartomizers