The Nikon 55-300, is not a fantastic lens, but, it is more than good enough for an awful lot, and it is relatively cheap, and from hint-imation of where you are at in your photography, I would suggest it's as good a place to start as any; buy 2n'd hand, if/when you outgrow it, move it on and buy something more suitable for intended use.... I am not a newby coming from film and, I haven't out-grow mine yet, in five years, and see no real need to upgrade it... Kit 18-55 on the other hand, is far more used, and would be far better place for me to put any up-grade spend....
Alternatively, and possibly a better bet; save even more pennies and buy the older 55-200 version. You can still get these new, from some places for under £100 if you hunt (I was looking out for one possibly for O/H, and recall spotting one new for £79!), and 2'nd hand for relative pennies, under-rated so much by simply not being a '300-class' lens, now that 18-200 super-zooms are rivaling their top end reach..... What you loose on the long end, is not a lot, and with the crop-factor of DX cameras these fit, anyway, is still the equivalent of a 300mm lens on full-frame / 35mm film, on which 200mm was about as long as zoom's got, before you had to look at prime tele's or mirror lenses; working the crop backwards, that is the same sort of framing as is maybe a 130 or 140m lens on crop.... so it is still a very useful amount of 'reach', and limit, likely to stop you racking all the way to the long end, consequently stopping down the aperture so much, making focus much more critical, and support and shutter speed so much more significant to avoid motion blurr.. so you will likely get more better pictures, even if the subject in them isn't so pominant in the frame..
Back-Ground 'blur', does not depend on the aperture... it depends on the camera to subject distance, then the depth of field about that focus distance, which is a % of the focus distance, and does change with the aperture; but most long lenses have further close focus distances, so the DoF beneath 'infinity' will tend to more often be in the 'blurry background' region... but DoF extends 1/3 in-front and 2/3 behind the distance you focus on, which if you turn 'off' the Auto-Focus need NOT be your actual 'subject'... manually focusing you can pull your focus distance ahead of your subject, still keep them in the DoF zone, and chuck back-ground oof far more and far earlier, as the shorter focus distance will shrink the DoF that is a % of it... it's a cheap trick... but of you like that blurry back-ground effect, then you would do far better learning the 'craft' and understanding what makes it, not hoping buying a new gadget will do it all for you..... coming from manual focus, and often manual exposure film camera's I find it utterly amazing that the "Po's Go-Manual' mantra is so often shouted, and applied almost solely to the metering system, not the focusing, that's left to a far less reliable red-dot! But still!
For nature and sports? Blurry back-grounds really should be the least of your worries. I don't do nature, but, skill there I am assured is in the stalking skills to actually get the creature in the frame to start with; if you have any luck in that, I wouldn't push it trying to get arty with the back-ground effects! Especially as above, if you use more 'reach' to get your creature filling the frame, rather than old fashioned getting close by getting close, patience using a hide or diligent and silent stalking, that extra 'reach' ad the inherent compromise of aperture and shutter speeds, and the issue of holding camera steady enough to avoid motion blur at such is often workng against you, and why, really dedicated nature whalla's use hides, tripods, and patience to let the little furry things get close to them, rather than a long lens or trying to creep silently through the woods and hold a camera steady hand holding! Sports? People playing soccer or whatever aren't much different, but you can usually get close by getting close, without them running away, and goal-posts, or finish lines tend not to go anywhere, pitches don't wander about or courses move, so even f the subject's a bit quick, there's only so far it ca go and you aught know where it's going to be or likely to go, which can make the stalking some-what easier.... And filling the frame with a face, may make a picture with a lot of 'impact'... but, if you want a photo of a face, is trying to grab it in the heat of action the best place to get it? Interest there is in the full scene, the bits you are throwing away with a tele-lens, that show the rest of the game, and give that face 'context' and reason, and explain that expression or pose or whatever. Make the wide 'work' for you... I have a 'set' shot two years back with the 55-300, at a classic motorbike scramble in Worcestershire; 1/4 mile of track and the tele-end of the zoom was only really useful for isolating a single bike on the start grid..... If wanted a picture of the rider straddling his bike, filling the frame, I would have got a much better shot, walking around the paddock when they weren't racing, and asking them to 'pose'! When I could have probably got a better posed portrait with the 18-55 or 'short' zoom! And more blurry back-ground if that was what I was after! don't think I shot very much at all, over 180mm, according to the exif, and I eventually left the 18-55 on the front, for how often I was running into the buffer at the wide end of the big-zoom, trying to get that 'context' onto the frame to show what they were at; especially when the leader had opened up a gap on his rivals, ad there was no-one else near to getting into the same frame with them, to show they were 'racing'!
O/H, with similar aspirations to you for nature, has bee frustrated, not getting frame filling sparrows ad what not...which I blame squarely on her twitching know how ad stalking skills, far more than the limitations of her 18-55 kit lens.... I don't know much about starlings and such... but I do know they like to wake me up when the sun starts shining, so at a guess, getting up early to catch them when they are so obviously about would be a good start, rather than going out (with the dog!?!) at lunch time! And do they migrate in autumn? So February when 'There's nothing else to do!' might not be the most likely time to find many subjects!! THIS, knowing your subject, is probably more important than your lens! But, lending her my 55-300... just as frustrating, and when she has come back with anything, trying to use 'too much zoom' and filling the frame with subject, the implications on the aperture and shutter speed, amplified by 'hand holding', she has tended to have even less success than she did in February with the kit! Hence looking out for the more moderate 55-200 for her....
The oft applauded and recommended Siggma 70-300, is a fantastic lens, well rated by so many sports and nature snappers, and oft suggested 'better' than the Nikon equivalent... though often on a DX small sensor body. It IS notionally a 'budget' full-frame lens, and on full frame, it's sometimes citasised for not being as sharp as it's reputation suggests; BUT birders seem to love it, and oft exploit the small sensor cameras 'crop factor' to get the equivalent reach of a 450mm lens on full-frame, only getting an image from the 'sweet-spot' from the middle of the lens, the small sensor 'crop' lopping off a lot of edge elaboration, so 'flattering' its technical performance....... new, they were, ISTR a £300 lens not long ago; but prices seem have dropped though, and they now rival the Nikon 55-300, and might be picked up new for around £160, or perhaps £100 2nd hand.. which does make it a very close contender to the Nikkor 55-300, and it is almost certainly a better lens... IF that extra reach at the long end REALLY is worth it, and the sacrifice at the wide end, where I personally find it more useful, not so consequential....
BUT, very very easy to go over-the-top with a long lens and have them start to make lie more work for you than less, especially if you are expecting them to deliver, when you have't learned the craft to give them the chance to deliver! There a more conservative zoom, can start to help you learn where more would be more useful, if not, at least help you avoid using too much and getting so many utter suffers i the process.
Which, puts that 55-200 back into very strong contention; half the price, new, of the Nikkor 55-300, or Sigma 70-300, with the added at the wide end over that; and an even bigger bargain 2nd hand, so many sold in kits, and upgraded for 'bigger better more' 300 class lenses; there are a lot around, and you might pick one up for under £40 with a little looking and a little haggling, IF 'cheap' is the major criteria here....
But beyond that, learning to look, not expecting the gear to do all the work for you; learning to know how to get better photo's, and now zooming 'out' to show the context and the bigger picture around your subject, rather than zooming in, and obsessing on just that subject, the lesser length of the lens, needn't be major impediment, certainly an advantage to upping your odds if you make newby mistake of racking to the max all the wile, but giving you more reason to learn to get what you want from craft, rather than kit. So it certainly has a place on the learning curve, even if it eventually serves its purposes and teaches you where and what 'better' lens you would really get most use and value from, along the way.
But.. stepping away from any purchase recommend at all.. shift of emphasis from kit to craft; you can do an AWFUL lot with not a lot. 18-55 kit are pretty restrictive by modern standards, but, back when all we had was film, cameras came, as standard, with a fixed 50mm prime on the front! (30mm equiv on DX Dgi) A 35-70 zoom, (23-46mm equiv) was a luxuary, and many 'telephoto' zooms were often only 70-140, (46-93mm equiv), and anything much over that, we were looking at dedicated 'prime' tele-photo's, of maybe 200 or 300mm, and often at rather compromised, fixed, slow aperture, elaboration prone prirne 'mirror' lenses, of maybe 300 or 500mm... but folk got photo's with such limited kit, learning the craft, and working within it's limitations; expecting to have to use craft over kit, to get what they wanted; and that often meant applying patience, knowing their subject, where and when they were likely to get it, and pttg themselves there to get it, not expecting to plonk themselves wherever, and have something i the gadget bag to do it for them. That still apples.. and works, and even if you have got better kit in the bag, craft will still help you get more still from it. Which begs the conclusion, IF budget is critical, don't spend, LEARN. S much on-line these days you don't even need go to a library or buy a book; just remember that a lot f resources propagate and perpetuate the 'sales myth' that you must have this or absolutely need that... 'cos the folk offering the resources make their living flogging it! Or are given it by the folk flogging it for the recommend other's do! so treat with seasoning, then go practice the craft.. THAT if you have limited budget is best place to spend it; getting photo's, not gadgets. Don't matter how many quids worth of glass or silicone have in the camera-bag.... if I want photo's have to take it some-where and put it to work... but that is where 'better' photo's will be found... out and about, seeng stuff worthy of a photo..... not in the camera shop!
