How the hell are you supposed to get a kill in this game? I've been shooting people left right and centre, getting the red cross to indicate a hit, but haven't registered a single kill, or even an assist yet.
Currently sitting at 0/11 K/D ratio.

I'm guessing it gets easier once you have access to better weapons.
I think a large part of it will be practice. Most people start out like that (myself included, that was just a long time ago and I'm by no means great so still learning and improving myself) but as you get to know the game and find a weapon you like it evens out.
Also remember some people have been playing for 18 months, so its often not that they're too powerful for you to kill in a conventional mmorpg sense, its just they have more experience and practice which will all come with time. Two things I would say is keep checking your minimap as it will save your life and get you kills, and secondly don't always rush towards the nearest control point or choke point, think about how you approach a fight and look for another angle.
The starter weapons are fine in most cases, although as NC I would avoid the default LMG the Heavy Assault has (Gauss SAW) until you have some more certs because it really needs some attachments to be of good use and the slow rate of fire hurts when in close combat. Yes some of the weapons you can buy with certs or station cash will be better in certain situations but these are more sidegrades than upgrades for example they might trade damage per bullet for higher rate of fire or have higher bullet damage but a smaller magazine. The default guns are by no means bad you just need to play to the weapons strengths and find one that suits you. (Try VR training to test all the guns).
If I were starting again as NC myself I'd play Medic to begin with. Assault rifles are generally good all round guns, you'll be useful to your teammates and you'll get plenty of certs both from killing and healing/reviving people. The default Gauss Rifle is a great gun, Aiming down sights is preferred but put a laser sight on it and you can hip fire reasonably well if needed making it a nice all round gun with decent damage and range.
Consider purchasing (with certs) a suit slot or two for the medic to begin with, this will depend on what you're dying to most, if explosions go flak, if bullets go nanoweave (note this doesn't help with headshots).
Get a few levels of the medic tool certed up (like all but Max rank), at Max rank it revives people at full health and is much faster, it is also a cert machine in big battles and very useful to your team. Also start certing your ability (Press F to activate) which is an Area of Effect Heal over time that also heals you, this may help you survive a bit.
Once you done that get Max rank of the medic tool and ability. Then you can look at adding stuff like C4 for your utility slot (useful for MAX units, vehicles etc), Revive Grenades or looking at another class.
Of course all of the above is just my opinion, if you want to play a different class to medic to start with then do so, it isn't going to hurt your progress.