As a beginner, the kit you have will almost certainly be 'better' than you are. The biggest limitations to most people's photography isn't the hardware; it's the 'imagination' to see a picture worth capturing; the dexterity to capture it, and the time ad opportunity to do so...
Hence the many camera widows, grumbling about how many cameras we all have knocking about, totally unsympathetic to why going out and walking around a stately home or something 'out of season' is more important than another visit to Dunelm to choose new wall-paper.... as we try and 'make' opportunity to use the kit we got, and find the inspiration to 'see' a picture worth capturing, and develop the dexterity to d so... you HAVE been warned.... but, practice-practice-practice is the only way, really, and money spent, getting out ad dong, is likely to do more for you than any amount of money spent on more 'kit' that'll take time away from using that bit you already have, as much as you can, as well as you can.
You're 18-55 'kit' lens is a very good bit of kit for the money; bundled with almost every DX camera Nikon sell, it benefits from a huge economy of scale to make it as good as they can for the money, that they just don't have in other,less popular lenses, and they have to, as its the lens thier camera is most likely goig to be judged on. The only real 'shortcoming' it has, as far as you need be concerned, is that it is, in compact camera terms only a 3x zoom... ie has a relatively short range between wide-angle and tele-photo..
Damn useful one though, and the range probably most used for most general photography; And, I would stab that few would really miss anything much wider than its 18 end, and an awful lot would do better missing anything much longer.... more 'magnification' a zoom lens gives, making something bigger in the picture, so the more critical 'framing' your shot tends to be, holding the camera rock steady, or using a support becomes and using fast enough shutter speeds, that any movement the frame or camera doesn't cause 'blurring' across the picture..
So while it might be a bit limited by that zoom range, that needn't be a draw-back; turn it to your advantage; recognise that its encouraging you to think 'how' to get the picture you want, rather than expecting to be able to twiddle a knob ad the camera just give it you, whether it's 'zoom with your feet' and get closer to the subject to make it larger in the picture, or to find a better angle to get more in the shot; and while its doing that; its also pretty tolerant of error and undemanding of expertise; helping you develop your 'eye', its also helping you avoid too may duffers...
PRIMES.. non-zoom lenses. As has bee said, twist the zoom ring to about the middle, on your 18-55 and leave it there; you have the same angle of coverage that a 35mm 'prime' would offer... actually, just had a quick look at mine and the 35mm focal length is actually marked on the ring! But on a prime, that's all you got; you cant go wider or nearer at the twist of a ring.
And as has bee said; 'usually' reason people buy primes, is that they have a 'special reason' for wanting one, where they have run out of versatility on the kit zoom, and a 'prime' lens, being simpler in construction, is often cheaper and easier to make, so the most cost effective way of getting whatever 'extra' is being sought. Be it a more 'wide angle' or more 'telephoto' or some idea of 'better optical resolution'.
Common, reason though is for a 'faster' aperture. The main difference between the kit 18-55 lens set to 35mm and the Nikon AF-S 35mm f1.8 DX, mentioned, is that number f1.8, which is the 'maximum aperture size'.. and on the 18-55, a quick check on mine suggests that the 'max' aperture at 35mm is f5. In short, this can, if set that far open, let more light through the lens, which can allow photos in lower light, or as mentioned to get greater control over 'Depth of Focus'..
Old adage... if you have to ask, you either don't need it, and/or you cant afford it!
So, as a 'type' primes MAY at some point be of interest or use to you; IF you ever get to a point that you are trying to take pictures that challenge the kit lenses 'all round' capability for which a more 'specialist' lens may be appropriate or essential to get what you are looking for... but just one feature of a lens whether its focal length can be varied; it's more likely that you will want wider or longer lenses, before you want better or faster ones, and either way, its all an 'expensive' triangle, balancing the features with needs and wants with money in your pocket!
And for now, as said, if you have any burning a hole; use it to go places; take photos, and find OUT what you can do with the thing for yourself, and where YOU feel the lens may be letting you down.. if at all.
As said, just picked mine up to check as I typed; its sat on the front of my camera, as it is the 'most used', and 'most useful' and for what it does and what it costs, I, certainly have little compulsion to chuck it away in favour of anything else in any hurry...
So if you heed advice and go do; most useful 'stuff' to add to your kit is likely to be more memory cards, so you can take more photos.. more spare battery packs, so you can stay out longer and take more photos..... and a tripod!
Seriously, top tip; for ten years, with digital SLR's still too pricey for my blood, and 'kids', making lugging film SLRs about and giving them and childs the attention they need, I have done most of my picture taking with a pretty cheap, 5Mpix 3x zoom compact; and a little tripod has bee FAR more use getting the most out of it or pushing its capability, than biting the bullet to get a DSLR has! Sure it COULD and possibly does, help me take better photo's, more easily.... BUT... I still have to get out and go places and see stuff worth taking pictures of to get'em.. and tripod STILL helps get the most out of it, before looking for 'more' in alternative lenses...
TL DR - youn eed more PICTURES.. not more 'stuff' in the camera bag.