How long have you had your current set up?
How often have you 'really' hit the buffers with the 18-55?
Do you 'really' need that much more 'reach', all that often?
I have just been having much the same debate with my O/H who insists she needs a longer lens for wild-life, and remains unconvinced that it would probably make more problems for her than it solves, as she struggles to hold the ruddy camera still, as it is, and last time she tried to take 'wild-life' photo's spent fifteen minutes, with my 55-300, at the wide end, trying to get the frame centered on the squirrel in the tree,twenty feet from her, whilst it was jumping up and down, yelling "Over HERE, mate!" at her! (well, not really, but may as well have been! Lol)
With a very tight angle of view, you do have to scan the scene very carefully to actually get and keep your subject in the frame, then with that amount of effective subject 'magnification' it does make the focus that much more critical, as the Depth of Field around the subject becomes that much shallower, and you need higher shutter speeds to control motion blur ad camera shake, which compete with usually slower max apertures on longer lenses, ALL working against, rather than for you.
That Is the nature and challenge of wild-life photography, BUT, more moderate 'zoom' and a little less ambition, is probably going to help you get started.
Was pointing out to O/H that the 'long lens' I used for quarter of a century on my film cameras was a 70-710, which lacking the crop-factor is equivalent of only a 45-140 zoom on crop-sensor digital; the lack of ultimate 'zoom' no great handicap to an awful lot of more general photography and I rarely had much if any need to go longer, using the 3x adapter in the gadget bag. On the electric-picture-maker, whilst I have the added 'reach' provided by a 55-300, (and no tele-converter to stretch it any), I rarely find I am banging the zoom up against the stop at the long end, and am far more often using that lens at the wider end, between 55 & 140mm settings.....I would have to be pretty keen on wild-life to be so concerned that I would get more use from the upper-reaches of a 70-300mm tele to justify sacrificing that more oft used bit at the wide-side.
I'm actually deliberating whether, for what I most oft use it for, and 18-140 super-zoom would be actually more useful, and whether the slightly more drool-worthy 18-200 would be the most sensible upgrade I could make, as a general purpose walk about lens, that would make both the 18-55 ad 55-300 pretty much redundant, and provide some slight quality gain over both.. Then a 70-300 or similar longer long lens, as a stand-alone may make more sense, with no loss on that most used General Purpose middle ground.. but that's my muse....
For where you are at and what you are about; question has to be whether a 70-300 is really going to be as much help to you as you hope....
As to the 50mm prime? I have to say I'm even more sanguine. I bought the 35mm version for my daughter, an i-phone junkie, when she started her O-Level photo course; she took to it like a duck to water, and exploited it very much. Its a cracking little lens, with the 'normal' angle of view on an APS-C sensor camera. BUT, it s a bit of a sidestep to the world of prime-lens photography, and a 'little' bit of a one trick dog as far as the shallow focus effects you can get from exploiting the widest aperture, which is itself a little bit of a placebo in many cases, and not such great shakes over using the 18-55 to best effect, to exploit Depth of Focus, where switching 'off' the auto-focus, to focus ahead of your subject and put the slightly larger DoF of a smaller aperture where you want it, rather than where a red-dot thinks it should be! Same can be said of the 50 prime, TBH which I also bought daughter, later when she started to do more considered portraiture for A-Level photography. They are very nice lenses, and yup, they do have a bit of image quality advantage, but otherwise, main benefit I would credt either with is mostly just the brighter view-finder! I use similar 'legacy' primes fro film cameras on my own DSLR, and yeah.. primes can be fun, but he 'vogue' is enormously over-rated, and the 50mm prime is even more over-vaunted, IMHO thanks to the legacy of when that was the 'standard' lens that came with most 35mm Film cameras, ad the consequent, low cost ad high availability of them for folk to experiment with on Digital.
Which is probably of very little help to you what so ever, other than to question BOTH purchase ideas. Let alone help you prioritize. But, of the two, as you already have a lens covering the 50mm angle of view, and could achieve the shallow focus effects that are it's main forte with it, employing just a little know-how; yeah, you probably don't need stick that one at the top of the list.70-300, then comes to the top, as you have already apparently decided.. but, again whether that lens is the best choice to 'help' you do more than you can at the moment, remains a little mutable; and the 55-200, which is even cheaper than the 35 or 50 primes, brand new, and an absolute bargain second hand, often less than £75, could, to my mind be a far better stepping stone.
The cheaper 55-200, has that very useful 'middle reach' above that of the 18-55, that is probably far more useful for general photography; and whilst it may not have the ultimate 'reach' of a 300 class lens, its still got a far bit; and that small limitation, is likely to make it easier to scan and track and get a subject in the frame, and hold it there, and give you more chance to get more 'better' less fuzzy shots, from that extra tolerance zone around the subject... and what the heck, you can still crop down in post-process to the same effective framing for small loss in pixel count....so, prove a more versatile, all-round useful lens for GP photo, and a better training tool for wild-life, that after you have acquired th sort of skill set needed for that genre, would let you actually exploit a longer lens to get the sort of results you aspire to, rather than likely more often frustrate you in the attempt.
BUT, that is to just chuck up just one alternative approach / suggestion / option IF you are to reconsider the choices you seem to have already made rather than merely wish us to support and endorse them... end of the day, its your money, your call, all we can do is offer advice. If that isn't useful to you.. carry on, buy the 70-300 first, I am sure you will be very pleased with it, and wont regret buying that before the 50 prime..