Old Words of Wisdom for you to ponder i the midst of your ethusiasm; The more you Magnify the Subject, the more you'll magnify your errors....Long lenses are a double edge-sword.
Standing back and trying to use a magnifying glass to 'see close', you are limiting the angles you can view from, and expecting the camera to show you what you want to see, not learning the 'craft' to get a close up by getting 'close up', and learnig to MAKE the camera show you what you want to see... as others have said, work on your technique, that is where better photo' come from, ot the cameras.
Back to long lenses a sec, and the matter of 'Magnification', subject and error... The lens is a 'Lever' for light. Principle of the lever is that the 'lever ratio' will reduce the force needed to move something, but by an amount proportional to how much 'less' it will move. Like Jacking up a car.. you'd struggle to lift a car off its wheels, but if you use a jack, you can lift it 'easily', BUT you have to put in more 'effort'; the 'lever' multiplying the force you can apply, but reducing the amount the car lifts, so you have to do a lot of 'jacking' to get the car up off its wheel... apply to camera; the 'leverage' of a longer lens, may make it 'easier' to 'get close' to your subject, but overall, it'll likely take more 'effort', or certainly more 'patience' to 'do the job'.. and lever's work both ways....
Now, hand-holding your camera, it can move. Any movement in the camera, when the shutter is open, will result in the subject 'streaking' or blurring in the frame... more magnification you have making that subject bigger in the frame, more 'blurr' you are likely to get, and the more 'obvious' it will be in your picture... worse, your longer lens.. well, its longer! Its hanging further off the front of the bit of the camera you are holding, so it has ore actual mechanical leverage working against your grip, AND it'll probably be heavier, adding to it; this tends to make 'support' using a tripod or mono-pod even more important, and that can make it more awkward to set up a shot, and more cumbersome to change; 'slowing you down' demanding that 'patience', AND the 'Field-Craft' to plan your pictures, predict where subjects are going to be and work around the limitations of NOT being able to just 'grab' shots from the hip, as they occur to you....
THIS is the start of 'Proper' photography, as opposed to taking 'Snap-Shots.. planning, pictures, setting up to take pictures, applying the patience to MAKE the situations, and make them appear AS you want to capture them.... not expecting merely to have a wonder camera that will do it all for you... Viration Reduction lenses are great, Auto-Focus is fantastic, BUT, skill is still BEHIND the camera, not IN it!
"Yeah Yeah Yeah! I know all this BUT I WANT a longer lens! Just TELL me what to buy!"
Oh-Kay.....
The Nikon 55-200 is a very useful 'starter' lens. A DX sensor camera like the D3200 (which is what I have), it gives 1.5x the magnification we used to get on 'full-frame' film cameras, where 210 was pretty much as 'long' as most folk would go, without getting into the rather compromised world of 'Mirror' lenses or VERY expensive 'long' primes; YUP it's NOT as 'long' as other suggested lenses.... this is NOT bad thing.. remember, more you magnify the subject, more you magnify any error! It's 'enough' to start getting you close-ER to your lanky-rabbits.. and its CHEAP. It's been pushed down the price list by the VRII series lenses and the 55-300, and with the 55-300 now available for around £150 new, the 'old' 55-200 is often offered for around £100, and you should be able to get one
second hand for not a lot more than half that, maybe £60 or so, from some-one who has upgraded to a 70-300 or more.. WHICH is what I would fully expect YOU to do at some point, if you pursue this... but, if you DO, then, even a 70-300, for all the technical superiority any-one may mention, is likely to still lack the 'reach' you crave.. while, still magnifying as much 'error' as subject. the 55-200, accepting the lesser magnification, gives you a chance to start developing the field craft, from the outset, NOT expecting the camera to do it all for you; with the small bonus, that you still have, at the short end, the 55mm wider angle of your kit lens..... means that while you are shooting from the hip, you can still 'rack out' to where you left off with that lens.
The
Nikon 55-300, as said, the 'newer' long kit; with better VR system and a bit more magnification, is the more 'superior' lens, and new, still very affordable if you shop around; but, technically 'superior' isn't always 'better', or 'better for me'. As sad, starting out YOU and your level of skill, will be the most limiting factor, not the camera or the bit of glass on the front, and as said, if yo take this further, ad you do develop your field craft, the eve the more superior lenses being suggested, are at some point, likely to become a bit limiting, and you'll want something 'more'.. so not over reaching, not over SPENDING (as has been warned, probably the most significant danger to this game!) Two suggested, offer plenty to be getting on with, the 55-200, the more for your money.
Yup; the Sigma & Tameron lenses are technically superior in many, many ways, but for where you are at, here and now, I just dont think what might make them 'superior' is likely to be of that much importance; it's like riding a bike; you don't need a club-competition, enthusiasts mountain bike to learn to ride! You just need something that has wheels and pedals, to get you started!
I think ill stick to your first suggestion 
